University  of  California. 

FROM   THE    LIBRARY   OF 

Dr.  JOSEPH   LeCONTE. 

GIFT  OF  MRS.   LECONTE. 
No. 

Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

l\?lfcrosoft  Corporation 


httb://www.archive.org/details/bourdonsarithmetOObourrich 


BOURDON'S 


ARITHMETIC: 


CONTAININQ 


A   DISCUSSION   OF   THE   THEORY   OP   NUMBERS. 


TRANSLATED  FROM  THE  FRENCH  OP  M.  BOURDON,  AND  ADAPTED 

TO  THE  USE  OF  THE  COLLEGES  AND  ACADEMIES 

OF  THE  UNITED  STATES, 


CHAELES   S.  YEI^TABLE, 

UOEHTUTE  INSTRUCTOR  IN  THE  UiaVERSITY  OP  VIRGINIA ;  FORMER  PROFESSOR  OP  MATHEMATICS 

IN  HAMPDEN   SIDNEY  COLLEGE,  VIR6INU ;   FORMER  PROFESSOR  OF  NATURAL 

PHILOSOPHY  AND  CHEMISTRY  IN  THE  UNIVERSITY  OP  QEORGU. 


PHILADELPHIA : 
J.  B.   LIPPINCOTT    &    CO. 

1858. 


V.'     ^ 


Entered,  according  to  Att  of  Congress,  in  the  year  1857,  by 

J.  B.  LIPPINCOTT    &    CO., 

in  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Eastern 

District  of  Pennsylvania. 


PREFACE 


I  AM  led  to  offer  the  present  translation  to  the  public, 
from  the  conviction  that  such  a  work  is  very  much  needed 
in  our  Academies  and  Colleges.  In  fact,  a  long  experience 
in  teaching  has  convinced  me  that,  one  great  difficulty 
which  the  young  student  has  to  encounter  in  the  study  of 
Algebra  and  the  higher  branches  of  analysis,  results  from 
the  want  of  sound  philosophical  ideas  on  the  fundamental 
properties  of  numbers,  and  from  the  fact  that  the  funda- 
mental operations  of  Arithmetic  are  generally  learned  by 
rote,  and  not  pursued  as  a  system  of  close  reasoning. 
Bourdon's  treatise  is  the  one  adopted  in  the  schedule  of 
public  instruction  by  the  University  of  France.  In  pre- 
paring the  translation,  I  have  compared  the  seventh  with 
the  twenty-ninth  Paris  edition,  and  endeavoured  to  select 
the  best  methods  of  each.  In  this  selection  and  arrange- 
ment, I  have  followed  the  outline  of  the  lectures  upon 
Arithmetic,  delivered  by  the  late  Professors  Bonny  castle 
and  Courtenay,  in  the  University  of  Virginia.  The  tables 
\have  been  re-arranged,  and  a  collection  of  examples  an- 
nexed to  the  work. 

The  portions  of  Bourdon's  very  complete  treatise  on 
the  Extraction  of  Roots,  Progressions,  Logarithms,  and 
their  applications,  I  have  left  out,  because  they  are  very 
thoroughly  discussed  in  the  best  treatises  on  Algebra 
adopted  by  our  Colleges  and  Universities.    I  have  followed 

(iii) 


IV  PREFACE. 

the  author  in  introducing  some  few  of  the  signs  and  pre- 
liminary definitions  of  Algebra.  This  usage  the  author 
well  defends,  as  follows :  —  "To  attempt  to  make  known 
even  some  of  the  simple  properties  of  numbers  without 
employing  the  signs  of  algebra,  is  to  present  them  in  a 
manner  very  incomplete  and  little  methodical.  To  use 
these  signs  to  some  extent,  enables  us  to  establish  the  con- 
nexion between  these  properties  and  their  most  important 
applications.  Moreover,  the  discussion  of  these  properties, 
a  knowledge  of  which  is  essential  to  a  thorough  knowledge 
of  arithmetic,  cannot  properly  enter  into  the  elements  of 
algebra,  without  breaking  the  chain  of  theories  which  con- 
stitute this  other  branch  of  mathematics.  In  fine,  the 
work  is  designed  for  those  who  wish  to  make  the  first  steps 
in  the  career  of  a  scientific  or  liberal  education  in  a  sure 
and  profitable  manner."  The  translator  hopes  the  present 
treatise  will  be  a  useful  addition  to  the  means  of  thorough 
instruction  in  the  United  States. 

C.  S.  V. 

LoNOwooD,  Va.,  1857. 


CONTENTS. 


PART  FIRST. 

.INTRODUCTION. 

Aeticles  1-9.  —  Preliminaries  —  Numeration,  spoken  and  vrritten — 
General  Ideas  on  Fractions,  and  the  Four  Fundamental  Opera- 
tions    Page  9 

CHAPTER  I. 

Of  the  Four  Fundamental  Operations  on  Entire  Numbers. 

Articles  10-41. — Addition — Subtraction — Multiplication — Division 
— Applications  and  Exercises  on  Chapter  First 17 

CHAPTER  11. 

Vulgar  Fractions. 

Articles  41-65. — Introduction — Reduction  of  Fractions  to  a  Com- 
mon Denominator — Of  the  Least  Common  Multiple — Simplifica- 
tion of  Fractions — Of  the  Greatest  Common  Divisor  of  Two 
Numbers — The  Four  Fundamental  Operations  upon  Fractions — 
Fractions  with  Fractional  Terms — Fractions  of  Fractions — Ap- 
proximative Valuation  of  Fractions — General  Observation  on 
the  Calculus  of  Fractions 59 

CHAPTER  III. 

Compound  Numbers. 

Articles  65-81. — Systems  of  Weights,  Coins,  and  Measures — Pre- 
liminary Operations — Compound  Numbers — Addition,  Subtrac- 
tion, Multiplication,  and  Division  of  Compound  Numbers — Exer- 
cises    85 

1*  (V) 


VI  CONTENTS. 

CHAPTER  IV. 
Theory  op  Decimal  Fractions — Decimal  System  of  Weights,  &c. 
Articles  81-109.  — Introduction  —  Use  of  the  Point  or  Comma  — 
Fundamental  Principles — Of  the  Four  Operations  on  Decimal 
Fractions — Valuation  of  the  Quotient  of  a  Division  in  Decimals 
—  Conversion  of  a  Vulgar  Fraction  into  Decimals  —  Decimal 
System  of  Weights  and  Measures — Divisions  of  Thermometers 
and  Circumference — Conclusion  of  Part  First — Exercises 100 


PART  SECOND. 

CHAPTER  V. 

General  Properties  of  Numbers. 

Articles  109-154. — Introduction  —  Use  of  Signs  and  Preliminary 
Definitions — Theory  of  Different  Systems  of  Numeration — Some 
General  Principles  of  Multiplication  and  Division — Divisibility 
of  Numbers — Verification  of  Multiplication  and  Division  by  the 
Properties  of  9  and  1 1 — All  the  Divisors  of  a  Number — Remarks 
upon  the  Greatest  Common  Divisor  of  two  Numbers  —  Prime 
Numbers — Greatest  Common  Divisor  of  Several  Numbers — Least 
Common  Multiple — Periodical  or  Repeating  Decimals — Some  of 
the  Properties  of  Periodical  Decimals — Exercises 128 

CHAPTER  VI. 

Op  Ratios  and  Proportions — Resolution  of  Questions  which  depend 
UPON  Proportional  Quantities. 

Articles  154-188.  —  Introduction — Ratios  and  Proportions  —  Their 
Principal  Properties  —  Equidifferences  —  Resolution  of  some 
Questions  in  Simple  Rule  of  Three  —  Of  Direct  and  Inverse 
Ratios — Employment  of  the  Method  of  Reduction  to  Unity  for 
all  Questions  of  Compound  Proportions — Remark  upon  the  Use 
of  Direct  and  Inverse  Ratios  in  the  Practical  Solution  of  these 
Questions — Rule  of  Simple  Interest — Rule  of  Discount — Rule 
of  Fellowship — Rule  of  Alligation — Some  Miscellaneous  Ques- 
tions— Exercises 179 

Collection  of  Examples  on  all  the  foregoing  Chapters 220 

j^ote  A. — Different  Systems  of  Numeration 237 

Note  B. — Abbreviated  Methods  of  Division  and  Multiplication 240 


Signs  made  use  op  in  the  Work. 

1st.   +  plus,  the  sign  of  addition. 

2d.    —  minus,  the  sign  of  subtraction. 

3d.    X  multiplied  hy,  the  sign  of  multiplication. 

4th.  -J-  divided  hy,        "         "      division. 

5th.  =  equal  to,  "         "      equality. 


(vii) 


"    O^  THE 

DIVERSITY 


ELEMENTS  OF  ARITHMETIC 


FIRST    PART 


INTRODUCTION. 

1.  We  call  magnitude,  or  quantity,  every  thing  which  admits 
of  increase  or  diminution.  For  example,  lines,  surfaces,  solids, 
intervals  of  time,  weights,  are  magnitudes.  We  can  only  form 
an  exact  idea  of  a  magnitude  by  comparing  it  with  another  mag- 
nitude of  the  same  species,  and  this  second  magnitude  is  called 
unity,  in  as  much  as  it  is  to  serve  as  a  term  of  comparison  for  all 
magnitudes  of  the  same  species.  Thus,  when  we  say  that  a 
wall  is  twenty  yards  long,  we  are  understood  to  have  already  ac- 
quired the  idea  of  the  unit  of  length  called  yard,  and  we  sup- 
pose that,  after  having  laid  down  the  yard  twenty  times  upon  the 
length  of  the  wall,  we  have  arrived  at  the  end. 

Unify,  in  mathematics,  is  then  a  magnitude  of  any  species 
whatever,  taken  arbitrarily  or  in  nature,  which  serves  as  a  term 
of  comparison  for  all  magnitudes  of  the  same  species.  Whence 
it  follows  that  there  are  as  many  species  of  units  as  of  magni- 
tudes. 

The  result  of  the  comparison  of  any  magnitude  whatever  with 
its  unit,  is  called  number.     A  number  is  called  entire  when  it  is 

(9) 


10  NUMERATION. 

the  assemblage  of  several  units  of  the  same  species  or  denomina- 
tion. Thus,  twenty  dollars,  thirty  pounds,  eight,  twelve,  fifteen 
units,  of  any  species  whatever,  are  entire  numbers. 

A  fraction  is  a  part  of  a  unit. 

A  fractional  or  mixed  number  is  an  assemblage  of  several 
units  of  the  same  denomination,  and  of  ^parts  of  this  unit. 

2.  When,  in  enunciating  a  number,  we  add  at  the  end  of 
that  number  the  species  of  magnitude  taken  for  the  unit,  the 
number  is  called  concrete.  Thus,  five  feet,  fifteen  hours,  six 
leagues,  are  concrete  numbers.  The  first  time  we  pronounce  a 
number,  the  only  sense  we  can  attach  to  it,  is  the  representing  to 
ourselves  a  unit  of  a  certain  denomination,  to  which  we  compare 
another  magnitude  of  the  same  denomination.  But,  by  degrees, 
the  mind,  which  accustoms  itself  to  abstractions,  represents  to 
itself  a  collection  of  any  like  objects,  of  which  each  one  is  unity. 
In  this  case  the  collection  is  called  an  abstract  number,  because, 
in  enunciating  it,  we  make  abstraction  of  the  species  of  unit  to 
which  we  refer  it.  It  is  in  this  last  light  that  we  are  to  consider 
numbers,  in  the  discussion  of  the  methods  relating  to  the  differ- 
ent operations  which  we  have  to  perform  upon  them,  if  we  wish 
to  establish  these  methods  so  as  to  be  able  to  apply  them  to  all 
possible  questions. 

NUMERATION. 

8.  The  first  researches  on  numbers  should  have,  necessarily, 
for  object,  the  giving  them  names  easy  to  retain ;  and,  as  there 
exists  an  infinity  of  numbers  (since  we  can  add  to  any  number 
whatever,  already  formed,  a  new  unit,  which  gives  rise  to  a  new 
number,  also  capable  of  being  augmented  by  unity),  it  is  neces- 
sary to  find  some  means  of  expressing  all  numbers  by  a  limited 
number  of  words,  combined  together  in  fit  manner.  Such  is  the 
object  of  spoken  numeration.  . 

Again,  each  one  of  the  words  which  enter  into  the  nomen- 
clature of  numbers  being  expressed  by  several  letters,  it  was 
found  necessary  to  invent  an  abridged  mode  of  writing  these 
words  and  their  combinations,  in  order  that  the  mind  might  be 


NUMERATION.  11 

able  to  seize  with  more  facility  the  reasonings  which  we  are 
obliged  to  make  upon  the  numbers.  This  is  the  object  of  written 
numeration,  which  consists  in  representing,  by  a  limited  number 
of  characters  or  ciphers,  the  numbers  enunciated  in  the  ordinary 
language. 

4.  Spoken  Numeration.  —  Though  the  nomenclature  of  entire 
numbers  is  known,  for  the  most  part,  to  the  young  men  for  whom 
these  elements  are  written,  we  think  it  best  to  give  a  succinct, 
yet  methodical  analysis  of  it ;  for,  the  numeration  which  is  adopted 
in  nearly  all  countries,  is  founded  upon  this  nomenclature. 

The  first  numbers  are,  one,  two,  three,  four,  Jive,  six,  seven, 
eight,  nine.  These  numbers  are  called  simple  units,  or,  units  of 
the  first  order.  Adding  a  new  unit  to  the  number  nine,  we  form 
the  number  ten,  which  we  regard  as  a  new  denomination,  or,  spe- 
cies of  unit  called  a  ten,  or,  a  unit  of  the  second  order.  We 
count  by  tens  in  the  same  manner  as  we  have  counted  by  simple 
units.  Thus,  we  say,  one  ten,  two  tens,  &c.,  &c. ;  ten,  twenty, 
thirty,  forty,  fifty ^  sixty,  &c.  Between  ten  and  twenty  there  are 
nine  other  numbers,  which  in  English  have  the  names,  eleven, 
twelve,  thirteen,  fourteen,  fifteen,  sixteen,  seventeen,  eighteen,  nine- 
teen ;  names  established  by  usage,  showing  by  their  derivation, 
the  addition  of  the  preceding  simple  units  successively  to  the 
unit  of  the  second  order. 

Between  twenty  and  thirty,  there  are  also  nine  numbers,  which 
are  enunciated,  twenty-one,  twenty-two,  &c.  And  thus  we  can 
enunciate  all  the  numbers  up  to  ninety-nine.  This  last  number, 
augmented  by  one,  gives  ten  tens,  or  the  number  one  hundred, 
which  we  regard  as  a  new  unit,  or  unit  of  the  third  order ;  and 
we  count  by  hundreds  as  we  have  counted  by  units  and  tens. 
Thus,  one  hundred,  two  hundred,  &c.  Placing  successively 
between  the  words  hundred  and  two  hundred,  two  hundred  and 
three  hundred,  eight  hundred  and  nine  hundred,  and,  after  nine 
hundred,  all  the  numbers  comprised  between  one  and  ninety- 
nine,  we  form  the  names  of  all  the  numbers,  from  one  hundred 
to  nine  hundred  and  ninety-nine.  We  can  see  that,  in  the  enun- 
ciation of  all  these  numbers,  we  have  employed  only  the  generic 


12  NUMERATION. 

terras,  one,  two,  three,  four,  five,  six,  seven,  eight,  nine,  ten,  hun- 
dred, and  words  easily  derivable  from  these. 

Adding  one  .to  the  number  nine  hundred  and  ninety-nine,  wc 
obtain  a  collection  of  ten  hundreds,  or  the  number  thousand, 
which  forms  the  unit  of  the  fourth  order.  Having  reached  this 
number  it  is  agreed,  in  order  not  to  multiply  words  too  much,  to 
regard  thousand  as  a  new  principal  unit,  before  the  name  of 
which  we  place  the  names  of  the  nine  hundred  and  ninety-nine 
first  numbers.  Thus,  we  say,  one  thousand,  two  thousand,  nine 
hundred  and  ninety-nine  thousand.  A  ten  thousand  forms  the 
unit  of  the  fifth  order ;  a  hundred  thousand  forms  the  unit  of 
the  sixth  order. 

Now,  placing  between  two  consecutive  numbers  of  the  denomi- 
nation thousand,  as  twenty  ihoviSdiud  and  twenty-one  thousand,  the 
names  of  all  the  numbers  of  lower  denomination  than  thousands, 
it  is  clear  that  we  can  thus  enunciate  all  the  numbers  up  to  nine 
hundred  and  ninety-nine  thousand,  nine  hundred  and  ninety- 
nine.  This  last  number,  augmented  by  one,  gives  ten  hundred 
thousand,  or,  a  thousand  thousand,  to  which  collection  the  name 
million  has  been  given ;  in  the  same  manner  the  collection  of 
thousand  millions  is  called  billions;  the  collection  of  thousand 
billions  is  called  trillions,  and  so  on  to  infinity. 

We  count  by  millions,  billions,  and  trillions,  as  we  have  counted 
by  thousands  ;  and  it  is  easy  to  see  that,  by  joining  to  the  generic 
words  indicated  above,  the  words  million,  billion,  trillion,  quatr- 
illion,  quintillion,  we  will  form  the  nomenclature  of  all  imagi- 
nable entire  numbers.  "VVe  observe,  in  order  to  terminate  this 
part  of  the  subject,  that  the  million  is  the  unit  of  the  seventh 
order,  ten  millions  are  units  of  the  eighth  order,  hundred  mil- 
lions, units  of  the  ninth  order. 

5.  Written  Numeration.  —  Though  the  above  nomenclature  is 
very  simple,  still  we  would  find  much  trouble  in  combining  toge- 
ther two  or  more  large  numbers,  unless  we  had  some  abridged 
mode  of  writing  them.  This  is  easily  arrived  at  by  reflecting  a 
little  upon  the  nomenclature.  We  observe  at  once,  that,  among 
the  words  employed  to  express  numbers,  the  one  part,  as  one,  ten', 


NtMERATION.  13 

hundredy  thousand,  ten  thousand,  &c.,  express  the  units  of  dif- 
ferent orders,  while  the  words,  one,  two,  three,  nine,  express 
how  many  times  each  of  these  sorts  of  units  enter  into  a  number. 
This  being  established,  if  we  agree  to  represent  the  first  nine 
numbers  by  the  characters  or  ciphers, 

123        4567        8        9 
one,  two,  three,  four,  jive,  six,  seven,  eight,  nine^ 

the  whole  difficulty  consists  in  finding  a  means  of  making  these 
ciphers  express  the  different  orders  of  unity  which  compose  the 
proposed  number.  Then,  establishing  this  principle  (purely  con- 
ventional), that  every  figure  placed  to  the  left  of  another,  expresses 
units  of  the  order  next  higher  to  those  of  the  other  figure,  or,  in 
other  words,  that  when  several  characters,  signifying  the  first  nine 
numbers,  are  written  one  after  anotJier,  then  the  first  figure  to  the 
right  expresses  simple  units,  the  next  on  the  left,  tens,  the  third 
figure  counting  from  right  to  left,  hundreds,  the  fourth,  thou- 
sands ;  it  is  easy  to  see  that,  in  general,  we  can  represent  all  num- 
bers by  the  aid  of  the  preceding  characters. 

Character  0.  —  While  this  is  true  in  general,  nevertheless, 
there  are  numbers  which  the  preceding  convention  fails  to  repre- 
sent, unless  we  agree  to  use  an  additional  character.  If  we  un- 
dertake to  write  in  figures  the  numbers,  ten,  twenty,  thirty,  &c., 
these  numbers  containing  no  simple  units,  we  are  compelled  to 
adopt  a  character  which  has  no  value  by  itself  but  which  serves 
to  hold  the  place  of  the  units  of  the  order  which  is  wanting  in 
the  number  enunciated.  This  cipher  is  0,  and  is  called  zero. 
By  the  aid  of  this  cipher,  the  numbers,  ten,  twenty,  &c.,  are 
expressed  by  10,  20,  30,  40,  &c. 

In  the  same  manner,  the  numbers,  one  hundred,  two  hundred, 
&c.,  which  contain  neither  simple  units  nor  tens,  are  written  thus : 

100,  200,  300. 

In  general,  the  zero  is  a  cipher  which  has  no  value  by  itself, 
but  which  we  employ  to  hold  the  place  of  the  different  orders  of 

9 


14  NUMERATION. 

unity  which  may  be  wanting  in  the  number  to  be  written.  The 
other  characters  are  called  significant  figures,  and  have  two 
values ',  the  one  we  call  absolute,  and  is  no  other  than  that  of  the 
figure  itself  considered  alone;  the  other,  we  call  relative,  which 
the  figure  acquires  according  to  the  place  which  it  occupies  to 
the  left  of  other  figures. 

Now,  if  we  reflect  that  every  number  is  composed  of  simple 
units,  of  tens,  of  hundreds,  &c.;  that  the  collection  of  units  of 
each  order  is  equal  to  nine;  that,  in  the  case  where  a  number  is 
deprived  of  certain  orders  of  units,  we  have  a  character  to  hold 
their  places,  we  will  see  at  once  that  there  is  no  entire  number 
which  cannot  be  expressed  by  the  aid  of  a  certain  combination 
of  the  ten  characters  :  — 

1,  2,  3,  4,  5,  6,  7,  8,  9,  0. 

Take  the  example,  thirty-six  billions,  five  hundred  millions, 
twenty  thousand,  four  hundred  and  seven. 

This  number  contains  seven  simple  units,  no  tens ;  four  hun- 
dreds, no  ones  of  thousands ;  two  tens  of  thousands,  no  hundreds 
of  thousands ;  no  ones  of  millions,  no  tens  of  millions,  five  hun- 
dreds of  millions ;  six  ones  of  billions,  and  three  tens  of  billions ; 
then  the  number  will  be  represented  by  365000  20  407. 

The  system  of  numeration  which  we  have  just  explained,  has 
received  the  name  of  the  decimal  system,  because  we  emploj'  ten 
figures  to  express  all  numbers.  Ten,  or  the  number  of  characters 
employed  is  called  the  base  of  the  system. 

6.  Let  us  make,  now,  an  important  observation  :  it  results  from 
the  nomenclature,  that  every  number  can  be  divided  into  hun- 
dreds, tens,  and  simple  units;  into  hundreds,  tens,  and  ones  of 
thousands ;  into  hundreds,  tens,  and  ones  of  millions,  etc. ;  that 
is  to  say,  into  sets  of  simple  units,  thousands,  millions,  &c.,  each 
set  expressed  by  three  figures,  except  the  last,  which  is  that  of 
the  units  of  the  highest  order,  and  which  cannot  have  more  than 
two  figures,  and  sometimes  contains  only  one.  When,  then,  we 
have  become  familiar  with  the  manner  of  writing  the  numbers  of 
three  figures,  it  is  sufiicient  to  write,  successively,  the  different 


NUMERATION.  15 

sets  one  to  the  left  of  the  other;  the  set  of  units,  the  set  of  thou- 
sands, the  set  of  millions,  &c.  We  can  even  commence  at  the 
left;  that  is  to  say,  write  first  the  set  of  the  units  of  the  highest 
denomination,  and,  to  the  right  of  this,  the  other  sets  in  the  order 
of  the  magnitude  of  their  units.  It  is  thus  that  we  ought  to  write 
a  number  dictated  in  ordinary  language.  But  it  is  necessary  to 
take  care  not  to  omit  the  zeros  destined  to  replace  the  orders  of  units 
which  are  wanting ;  and  there  can  never  be  any  difficulty,  since  we 
know  that  each  set,  except  the  first  to  the  left,  must  always  con- 
tain three  figures.  Suppose,  for  example,  that  we  have  to  write, 
by  aid  of  our  characters,  four  hundred  and  six  billions^  twenty- 
eight  millions^  two  hundred  and  fifty  thousandj  and  forty-eight. 
Write  in  succession,  each  to  the  right  of  the  other,  the  period 
of  hill  ions ;  the  period  of  millions;  the  period  of  thousands; 
and,\sist\j,  thsit  of  simple  units  ;  we  will  have  406,  028,  250,  048. 

7.  It  is  upon  the  preceding  observation  that  the  following 
means  of  translating  into  ordinary  language,  any  number  what- 
ever written  in  figures,  is  founded.  • 

After  having  separated  the  number  into  periods  of  three  figures 
each,  commencing  at  the  right,  enunciate  successively  each  period ^ 
setting  out  from  the  first  period  on  the  left,  and  taking  care  to 
give  to  each  period  the  name  which  belongs  to  it. 

Example:  70345601.  This  number,  being  divided  70,345,601, 
is  composed  of  seventy  millions,  three  hundred  and  forty-five 
thousand,  six  hundred  and  one. 

8.  It  remains  for  us  still,  in  order  to  complete  the  theory  of 
enumeration,  to  show  the  mode  of  writing  fractions  by  means  of 
figures.  But  we  must  first  give  a  clear  and  precise  idea  of  frac- 
tions, such  as  we  consider  them  in  arithmetic. 

Let  us  suppose  that  we  have  to  determine  the  length  of  a  piece 
of  cloth.  Taking  the  unit  of  length  called  yard,  and  applying 
it  as  many  times  as  possible  to  the  length  of  the  piece,  two  cases 
may  occur,  either,  after  the  unit  has  been  applied  a  certain  number 
of  times — 15  times,  for  example,  nothing  will  remain — or,  we  will 
obtain  a  remainder  less  than  the  yard.  In  the  first  case,  the  piece 
will  contain  an  entire  numher  of  yards.  In  the  second  case,  it  will 


16  NUMERATION. 

be  necessary  in  order  to  have  the  whole  length  of  the  piece,  to  add 
to  these  15  yards  the  fraction  or  part  of  the  yard  which  remains. 
But  how  value  this  part  ?  how  compare  it  to  the  unit  ?  We  can 
first  conceive  this  unit  separated  into  two  equal  parts  or  halves; 
and,  if  the  remainder  is  exactly  equal  to  one  of  these  halves,  we 
say  that  the  piece  of  cloth  is  15  yards  and  one  half  long. 

If  the  remainder  is  less  or  greater  than  a  half^  yard,  we  con- 
ceive this  half  divided  into  two  new  equal  parts,  called  quarters. 

Instead  of  dividing  the  unit  into  two  or  four  equal  parts,  we 
can  conceive  it  to  be  divided  into  three  equal  parts  called  thirdsj 
&c.,  &c. 

Whence,  we  see  that  in  order  to  form  a  clear  idea  of  a  fraction 
of  a  unit  of  any  denomination  whatever,  it  is  necessary  to  con- 
ceive that  this  unit  be  divided  into  a  certain  entire  number  of 
equal  parts,  and  that  we  take  one,  two,  three,  &c.,  of  these  parts; 
these  parts  thus  taken,  constitute  what  is  called  vl  fraction.  Thus, 
the  enunciation  of  a  fraction  involves  necessarily  two  entire 
numbers,  to  wit:  —  th^t  which  denotes  into  how  many  parts  the 
unit  has  lieen  divided,  called  the  denominator ;  and  that  ichich 
denotes  how  many  of  these  parts  are  necessary  to  form  the  frac- 
tion, called  the  numerator.  For  example,  five-eighths  of  a  yard, 
thirteen-twentieths  of  a  pound,  are  fractions.  In  the  first,  we 
conceive  the  yard  divided  into  eight  parts,  and  that  we  take  five 
of  these  parts  to  form  the  fraction,  eight  is  the  denominator,  and 
five  the  numerator.  .  .  .  (We  see  that  in  the  spoken  numeration 
of  fractions  the  numerator  remains  unchanged  in  name,  while 
the  denominator  is  generally  changed  by  the  addition  of  th.) 

It  results,  also,  from  the  above,  that  a  fraction  is  a  magnitude 
referred  to  a  part  of  the  principal  unit,  which  part  we  can  con- 
sider itself  as  a  particular  species  of  unit.  Thus,  the  fraction 
thirteen-twentieths  of  a  yard,  being  composed  of  thirteen  times 
the  twentieth  of  a  yard,  this  twentieth  is  a  particular  unit,  which 
the  proposed  fraction  contains  thirteen  times.  This  being  estab- 
lished, two  fractions  are  said  to  be  of  the  same  species  when  their 
denominator  is  the  same,  (the  original  or  compound  unit  being 
likewise  the  same).    For  example,  five-twelfths  and  seven-twelfths 


ADDITION.  17 

of  a  yard  are  fractions  of  the  same  species ;  but  three-fourths  and 
two-thirds  of  a  pound  are  fractions  of  different  species  or  deno- 
minations, because  the  denominators  are  different. 

In  order  to  express  a  fraction  in  figures,  we  place  the  numera- 
tor above  the  denominator,  with  a  line  between.  Thus,  the  frac- 
tion three-fourths  is  denoted  by  |,  seven-twelfths  by  j'^^- 

Reciprocally,  |,  ||,  represent  the  fractions,  seven-eighths, 
thirteenth-fifteenths,  that  is  to  say,  we  enunciate  the  numerator, 
and  then  the  denominator,  and  add  the  termination  —  th,  to  the 
latter. 

CHAPTER  I. 

9.  Arithmetic  has,  for  it  special  object,  to  establish  fixed  and 
and  certain  rules  for  performing  all  possible  operations  upon 
numbers.  It  embraces,  besides,  the  study  of  a  great  number  of 
properties  which  have  been  discovered  during  the  researches  made 
in  order  to  arrive  at  these  methods,  or  to  facilitate  the  use  of  them. 
We  will  now  explain  these  operations  in  their  order,  recollecting 
that,  in  order  to  render  the  methods  independent  of  every  sort  of 
question,  it  is  best  to  consider  the  numbers  as  abstract  numhers. 

Nevertheless,  in  the  applications  designed  to  familiarize  begin- 
ners with  the  methods,  we  can  propose  questions  also  relating  to 
concrete  or  denominate  numhers. 

OPERATIONS   ON  ENTIRE  NUMBERS. 
ADDITION. 

10.  To  add  or  sum  up  several  numbers^  is  to  unite  all  these 
•numhers  into  a  sinylc  one;  or,  to  form  a  number  which  contains 
in  itself  alone  as  many  units  as  there  are  in  the  different  numbers 
taken  separately. 

The  result  of  this  operation  is  called  the  sum,  or  total.  The 
addition  of  numbers  of  a  single  figure  offers  no  difficulty.     It  is 


18  ADDITION. 

done  unit  by  unit.  Children  learn  thus  to  make  these  additions 
by  means  of  their  fingers,  and  fix  the  results  in  their  memory. 
In  this  way,  for  example,  they  find  thirty  to  be  the  sum  of 
5,  7,  8,  4,  and  6;  or,   that  42  is   the   sum  of  the   numbers 

7,9,6,5,8,7. 

OPERATION. 

Let  now  the  numbers  to  he  added  be  7,453         7,453 
and  1,534.  1,534 

After  having  written  the  numbers,  one  under  8,987 
another,  with  a  line  under  them,  we  commence 
with  the  simple  units,  and  say,  3  and  4  make  7,  which  we  place 
under  the  units.  Passing  to  the  tens,  5  and  3  make  8,  which  we 
write  under  the  tens.  Then,  4  and  5  make  9,  which  we  write 
under  the  hundreds.  Lastly,  7  and  1  make  8,  which  we  write 
in  the  column  of  thousands. 

The  number,  8,987,  found  by  this  operation,  is  the  sum  of 
the  two  given  numbers,  since  it  contains  their  units,  their  tens, 
their  hundreds,  and  their  thousands,  which  we  have  summed  up 
successively. 

OPERATION. 

Again,  let  it  be  proposed  to  add  the  four  num-         5,047 
bers,^  5,047,  859,  3,507,  846.     We  write  them  859 

one  under  the  other,  and,  commencing  with  the         3,507 
units,  7  and  9  make  16,  and  7  make  23,  and  6  846 

make  29.  We  place  the  nine  simple  units  under  10,259 
the  first  column,  and  retain  the  two  tens,  in  order 
to  add  them  to  the  figures  of  the  next  column,  which  are  also 
tens.  Passing  to  the  next  column,  we  say  that  the  two  reserved, 
and  4  make  6,  and  5  make  11,  and  0  make  11,  and  4  make  15. 
We  write  the  5  in  the  column  of  tens,  and  retain  the  1  hundred 
which  we  carry  to  the  column  of  hundreds.  Operating  upon 
this  column,  as  upon  the  preceding,  we  find  22  hundreds,  or  2 
hundreds,  which  we  write  under  the  hundreds,  and  2  thousands, 
which  we  retain  in  order  to  carry  them  to  the  column  of  thou- 
sands.    Lastly,  2  reserved,  and  5  make  7,  and  3  make  10.     We 


SUBTRACTION.  19 

place  the  0  under  the  thousands  and  advance  the  1  to  the  left, 
which  gives  10,259  for  the  required  sum. 

General  Rule.  —  In  order  to  add  several  numbers  together, 
commence  by  writing  them  one  under  another,  so  that  the  units 
of  the  same  order  may  be  in  the  same  column.  Add  then 
successively  the  figures  which  compose  each  one  of  the  vertical 
columns,  commencing  with  the  column  of  simple  units,  passing 
to  the  columns  which  are  on  the  left :  write  below  the  line 
the  sum  of  the  figures  of  each  column,  provided  the  sum  is 
expressed  by  a  single  figure.  But  if  it  exceeds  9,  in  which 
case  it  is  expressed  by  several  figures,  of  which  the  last  to  the 
right  represents  the  units  of  this  column,  and  the  others  to  the 
left  tens  of  the  same  order,  write  only  the  figure  of  units  below 
the  column,  and  reserve  the  tens  in  order  to  add  them  to  the  figures 
of  the  column  immediately  to  the  left.  When  you  have  operated 
in  this  manner  upon  all  the  columns,  you  loill  obtain  the  sum 
required,  because  it  results  from  the  union  of  the  units,  tens, 
hundreds,  &c.,  which  enter  into  the  given  numbers. 

11.  Remark. — If  the  sum  of  the  figures  in  each  column  does 
not  exceed  nine,  we  could  commence  the  operation  equally  well 
by  the  addition  of  the  units  of  the  highest  order  as  by  the  addi- 
tion of  the  simple  units.  But  as  it  happens  oftenest  that  several 
of  these  sums  exceed  nine,  if  we  commence  on  the  left,  we  will 
often  be  obliged  to  return  upon  our  steps,  in  order  to  correct  a 
figure  already  written,  and  increase  it  by  as  many  units  as  we 
shall  have  obtained  from  the  tens  of  the  following  column  in 
operating  upon  that  column.  For  this  reason  it  is  best  in  all 
cases  to  commence  on  the  right  rather  than  on  the  left. 

SUBTKACTION. 

12.  To  subtract  one  number  from  another  is  to  seeh  the  excess 
of  the  greater  number  over  the  less.  The  result  of  this  operation 
is  called  remainder,  excess,  or  difference.  So  long  as  the  numbers 
proposed  consist  only  of  a  single  figure,  the  subtraction  is  easy 


20  SUBTRACTION. 

Thus,  the  difference  between  9  and  6  is  3.  "VVe  can  easily  sub- 
tract a  number  of  a  single  figure  from  a  number  which  does  not 
exceed  twenty.  Thus,  take  7  from  13,  there  remains  6,  since  by 
what  we  have  learned  in  addition,  7  and  6  make  13.  In  the 
same  manner,  9  from  17  there  remains  8,  because  8  and  9  make 
17.  These  operations,  which  suppose  only  the  exercise  of  the 
memory  upon  the  addition  of  numbers  of  a  single  figure,  serve 
as  a  basis  for  the  subtraction  of  numbers  of  several  figures. 

Let  it  he  required  to  subtract  5467  from  8789. 

OPERATION. 

After  having  placed  the  smaller  number  under  the       8789 
greater,   and    underlined    the   whole,    we    say,    com-       5467 
mencing  with  the  simple  units,  7  from  9  leave  2,  which       3322 
we  place  in  the  column  of  simple  units;  passing  to  the     • 
tens,  6  from  8  leave  2,  which  we  write  in  the  column  of  tens ; 
the  same  operation  finally  upon  the  hundreds  and  thousands,  4 
from  7  leave  3,  and  5  from  8  leave  3,  gives  3322  for  the  re- 
quired  remainder.     For  by  the  nature  of  the  operations  which 
have  just  been  performed,  we  see  that  the  greater  number  con- 
tains more  than  the  second,  2  simple  units,  2  tens,  3  hundreds, 
3  thousands,  and  consequently  exceeds  the  smaller  by  3322. 

Let  us  propose  for  a  second  example,  to  find  the  difference 
which  exists  between  the  two  numbers,  83456  and  28784. 

OPERATION. 

Having  arranged  the  numbers  as  in  the  preceding       83456 
example,  we  say,  first,  4  from  6  leave  2,  which  we       28784 
write  under  the  units.     But  when  we  pass  to  the        54672 
column  of  tens,  we  meet  with  a  difficulty :  the  lower 
figure,  8,  is  greater  than  the  upper  one,  5,  and  consequently 
cannot  be  subtracted.     In  order  to  overcome  this  difficulty,  we 
borrow  mentally  from  the  hundreds  figure   1   hundred,  which 
equals  10  tens,  and  add  it  to  the  5  tens  which  we  have  already, 
giving  us  15  tens;  we  then  say,  8  from  15  leave  7,  which  we 
write  in  the  column  of  tens.     Passing  to  the  column  of  hundreds, 
we  observe  that  the  upper  figure,  4,  ought  to  be  diminished  by  1, 


SUBTRACTION.  21 

since  we  have  borrowed  this  unit  in  the  preceding  subtraction ; 
we  say,  then,  7  from  3,  which  is  impossible ;  but  we  borrow,  as 
before,  1  thousand,  which  equals  ten  hundreds,  giving  13  hun- 
dreds, and  take  7  from  13,  which  gives  6,  to  be  written  in  the 
column  of  hundreds.  Passing  to  the  thousands,  8  cannot  be 
taken  from  2 ;  but  8  from  12  leave  4,  to  be  written  in  the  column 
of  thousands.  Lastly,  as  the  figure  8,  of  tens  of  thousands,  on 
account  of  the  1  just  borrowed,  ought  to  be  replaced  by  7,  we 
say,  2  from  7  leave  5.  Thus,  the  remamdevj  or  the  excess  of 
the  greater  number  over  the  less,  is  54672. 

In  order  to  understand  how,  by  this  means,  we  arrive  at  the 
end  proposed,  it  is  sufficient  to  remark  that,  according  to  the 
artifices  employed  tft  effect  the  partial  subtractions,  we  can  ar- 
range the  two  numbers  in  the  following  manner : — 

Tens  of  thousands,  thousands,  hundreds,    tens,    units. 

1st  number,  7  12  13         15      6 

2d  number,  _2 8  7  8      4 

5  4  6  7      2 

From  this  we  see  that  the  upper  number  exceeds  the  lower 
one  by  two  units,  7  tens,  6  hundreds,  4  thousands,  and  5  tens 
of  thousands  —  or  exceeds  it  54672  units. 

Let  it  he  proposed,  for  example,  to  subtract  158429  from 
300405. 

OPERATION. 
99      9 

As  9,  the  units  figure  of  the  lower  number,  is  300405 
larger  than  5,  the  corresponding  figure  of  the  greater,  158429 
we  have  to  borrow  1  ten  from  the  first  figure  to  the  141976 
left ;  but  this  figure  being  0,  it  is  necessary  to  have 
recourse  to  the  figure  4,  of  hundreds,  from  which  we  borrow  1, 
which  equals  10  tens;  and  since  we  have  need  of  only  a  single 
ten,  we  leave  9  of  them  above  the  0;  we  then  add  1  ten  to  5, 
which  gives  15,  and  say,  9  from  15  leave  6,  which  we  write 
under  the  units.     Passing  to  the  tens,  we  say,  2  from  9  leave  7. 

For  the  hundreds,  as  the  upper  figure,  4,  has  been  diminished 


22  SUBTRACTION. 

by  the  1  whicli  we  borrowed,  and  as  we  cannot  take  4  from  3, 
we  have  recourse  to  the  next  figure  to  the  left ;  but  that  and  the 
figure  which  is  to  its  left  being  zeros,  we  borrow  1  from  the  next 
significant  figure,  3.  This  1  equals  10  of  the  order  following,  and 
100  units  of  the  order  thousands ;  and  since  we  have  need  of 
only  1  unit  of  this  order,  we  leave  99  of  them,  which  we  place 
above  the  two  zeros;  adding  1  thousand  to  the  3  hundreds,  it 
becomes  13  hundreds,  and  we  say,  4  from  13  leave  9,  which  we 
place  under  the  column  of  hundreds. 

In  the  two  following  subtractions,  each  one  of  the  zeros  being 
replaced  by  a  9,  we  say,  8  from  9  leave  1,  and  5  from  9  leave  4. 
Passing  to  the  first  column  to  the  left,  we  say,  1  from  2  (for  the 
figure  3  is  diminished  by  1)  leaves  1.  Thus  we  have  for  the 
required  remainder  141976. 

If  we  reflect  upon  the  manner  in  which  the  greater  number 
has  been  decomposed,  we  can  arrange  the  operation  thus : — 

hundreds  of  thous.,  tens  of  thous.,  thous.,  hundreds,    tens,    units. 

1st  number,  2  9  9         13  9      15 

2d  number,         _1 5  8  4  2        9 

1  4  19  7        6 

Then  the  greater  number  exceeds  the  less  by  6  units,  7  tens, 
9  hundreds,  1  thousand,  4  tens  of  thousands,  1  hundred  thou- 
sand, or  by  141976. 

General  Rule. — In  order  to  perform  the  subtraction  of  two 
numbers,  place  the  less  number  under  the  greater,  so  that  the 
units  of  the  same  denomination  fall  in  the  same  column  ;  then 
underline  the  two  numbers;  sid)tract  then  successively,  units  from, 
units,  tens  from  lens,  hundreds  from  hundreds,  &c.,  and  write  the 
partial  remainder's  one  to  the  left  of  another;  the  number 
formed  by  these  remainders  is  the  total  remainder,  or  the  result 
required. 

When  a  figure  of  the  lower  line  is  greater  than  the  figure  above 
it,  augment  mentally  this  last  figure  by  \(}  units,  and  diminish 
the  figure  to  the  left  of  it  by  one  unit. 


SUBTRACTION.  28 

Jf  immediately  to  the  left  of  an  upper  figure  less  than  the  one 
below,  corresponding,  there  are  one  or  more  zeros,  increase  this 
figure  above  mentally  always  by  10  units  ;  but  in  the  following 
subtractions  replace  the  Os  by  9s,  and  diminish  by  a  unit  the 
upper  significant  figure  which  is  immediately  to  the  left  of  these 
zeros. 

13.  First  Remark.  —  If  each  one  of  the  figures  of  the  lower 
number  is  less  than  the  corresponding  figure  of  the  greater,  we 
could  commence  the  operation  indifi'erently  at  the  right  or  left. 
But  as  it  often  happens  that  one  of  the  figures  of  the  less  num- 
ber exceeds  the  figure  of  the  greater  above  it,  the  partial  subtrac- 
tion cannot  be  effected  without  borrowing  from  one  of  the  figures 
to  the  left  of  that  one  with  which  we  are  operating;  for  this 
reason  it  is  necessary  to  commence  on  the  right,  in  order  to  bor- 
row when  there  is  need  of  it. 

14.  Second  Remark.  —  It  is  clear  that  instead  of  diminishing 
by  one  unit  the  figure  from  which  we  have  borrowed  it,  we  can 
leave  this  figure  unchanged,  provided  we  augment  the  corres- 
ponding figure  below  by  one  unit.  This  manner  of  operating  is 
in  general  more  convenient  in  practice. 

Thus,  in  the  ^ast  example,  after  having  said  for  the  simple 
units,  7  from  11  leave  4,  instead  of  saying  for  the  tens,  8  from  9 
leave  1,  we  say,  9  from  10  leave  1 ;  in  the  same  manner,  instead 
of  saying  for  the  hundreds,  7  from  13  leave  6,  we  say,  8  from  14 
leave  6,  and  so  on  for  the  rest. 

But  when  we  employ  this  modification,  we  must  be  careful  to 
augment  the  lower  figure  only  when  we  have  been  obliged  to 
borrow  in  the  subtraction  of  the  preceding  figures.  This  modifi- 
cation is  used  particularly  in  division. 

VERIFICATION  OF  ADDITION  AND  SUBTRACTION. 

15.  We  call  the  verification  of  an  arithmetical  operation, 
another  operation  which  we  perform  in  order  to  convince  our- 
selves of  the  accuracy  of  the  first. 


24  SUBTRACTION. 

The  verification  of  addition  is  effected  by  adding  anew,  but 
commencing  at  the  left  hand.  After  having  formed  the  sum  of 
the  figures  in  the  first  column  on  the  left^  we  subtract  it  from, 
that  part  which  answers  to  it  in  the  sum  total ;  we  write  down 
the  remainder,  which  we  reduce  mentally  into  units  of  the  order 
of  the  following  figure,  in  order  to  join  them  to  the  units  of  this 
order  in  the  sum  total.  In  the  same  manner  we  sum  up  the 
second  column  on  the  left,  and  subtract  this  partial  sum  from  the 
corresponding  part  of  the  sum  total ;  ice  continue  this  operation 
to  the  last  column  ;  the  last  subtraction  leaves  no  remainder. 

Thus,  after  having  found  that  the  four  numbers, 

6047 

859 

3507 

846 

have  for  their  sum 10259 

in  order  to  verify  the  result 2120 

we  add  the  same  numbers  commencing  on  the  left.  We  say,  5 
and  3  make  8  thousands,  which  we  subtract  from  10  thousands, 
leaving  2  thousands  for  remainder;  which,  with  the  figure  2 
hundreds,  make  22  hundreds ;  then  8  and  5  make  13,  and  8 
make  21,  which  we  take  from  22,  which  gives  for  remainder  1 
hundred,  which,  joined  to  5  tens,  forms  15  tens ;  4*and  5  make  9, 
and  4  make  13 ;  13  from  15,  there  remains  2,  which,  joined 
to  the  9  units  following,  gives  us  29;  lastly,  7  and  9,  and  7  and 
6  make  29 ;  29  from  29  and  nothing  remains ;  then  the  operation 
is  exact. 

The  verification  of  subtraction  is  effected  by  adding  to  the 
smaller  number  the  remainder  found  by  the  operation  ;  and  it 
is  evident  that  we  ought  thus  to  reproduce  the  greater  number, 
since  this  remainder  is  nothing  more  than  the  excess  of  the 
greater  number  over  the  less. 


SUBTRACTION. 

Thus,  in  the  annexed  examples,  after  having 
found  that  54682  is  the  excess  of  the  greater 
number  over  the  less,  if  we  add  this  excess  to 
the  number  28784,  we  ought  to  obtain  the 
number  83466  —  which  we  do  in  fact  obtain. 


25 

83466 

,  28784 

Rem.    54682 
Proo/83466 


16.  Here  we  give  some  examples  of  addition  and  subtraction, 
with  their  verifications. 


Additions. 

83054 

700548 

256870 

897597 

748759 

6588 

90874 

69764 

130909 

407300 

8746 

987846 

1319212 

1207047 

2:^^^0 

Subtractions. 

4276690 

4073050062 

20004001003 

2803767086 

' 

8405128605 

1269282976 

11598872398 

4073050062 

20004001003 

Problem.  — A  banker  had  in  his  chest  a  sum  o/*  $65, 750;  he 
gave  one  person  $13,259 ;  to  a  second,  $18,704 ;  to  a  third, 
$22,050 ;  to  a  fourth,  $9850  j  what  was  the  state  of  his  chest  after 
these  payments  ? 


Solution.  —  After  having  summed  up  the  four  sums  succes- 
sively paid,  we  subtract  the  sum  total  from  that  which  he  had, 
3 


26  MULTIPLICATION. 

and  the  result  of  the  subtraction  will  be  what  ought  to  remain  in 
his  chest.     Thus, 

13259  65750 

18704  63863 


22050  $1887  what  he  has  left. 

9850 


63863 

We  remark,  that  in  effecting  the  preceding  addition  and  sub- 
traction, we  have  considered  the  given  numbers  as  abstract,  al- 
thouoh  they  were  denominate  numbers  according  to  the  enuncia- 
tion of  the  question ;  but,  arrived  at  the  result,  1887,  we  have 
given  it  the  name  of  the  species  of  unit  which  the  numbers 
expressed  in  the  enunciation.  We  must  always  perform  the 
operations  in  this  manner,  when  we  wish  to  apply  the  results 
of  the  operations  to  questions  in  denominate  numbers.  ■  The 
results  being  altogether  independent  of  the  nature  of  the  nuin- 
bers,  we  consider  them  in  a  point  of  view  purely  abstract,  except 
in  giving  to  the  final  result  the  name  of  the  unit  which  the 
enunciation  of  the  question  indicates. 

MULTIPLICATIOlf. 

17.  To  multiply  one  number  hy  another ,  is  to  compomid  a 
third  number  with  the  first,  as  the  second  is  compounded  with 
unity.  Then,  if  the  two  given  numbers  are  entire  numbers, 
to  multiply  them  is,  to  take  the  first  as  many  times  as  there  arc 
units  in  the  second. 

We  call  the  result  of  multiplication,  product;  the  number  to 
be  multiplied,  multiplicand ;  and  the  number  by  which  we  mul- 
tiply, multiplier  ;  which  denotes  how  many  times  we  are  to  take 
the  first.  The  two  numbers  bear  jointly  the  name  of  factors  of 
the  product.  Properly  speaking,  multiplication  is  nothing  else 
than  addition ;  for,  in  order  to  obtain  the  result,  it  would  suffice 
to  write  the  multiplicand  as  many  times  as  there  are  units  in  the 


MULTIPLICATION.  Z  / 

multiplier,  and  then  add  all  these  numbers  together.  But  this  man- 
ner of  operating  would  be  very  long,  if  the  multiplier  was  composed 
of  several  figures ;  we  are  then  to  seek  a  method  of  simplifying 
it,  and  it  is  in  this  abbreviation  that  multiplication  consists. 

18.  As  long  as  the  two  factors  are  expressed,  each  one  by  a 
single  figure,  their  product  is  obtained  by  the  successive  addition 
of  the  multiplicand  to  itself;  thus,  in  order  to  multiply  7  by  5, 
we  say,  7  and  7  make  14,  and  7  make  21,  and  7  make  28,  and  7 
make  35 ;  this  last  number  being  the  result  of  the  addition  of 
five  numbers  equal  to  7,  expresses  the  product  of  7  by  5. 

Beginners  will  do  well  to  exercise  themselves  in  this  sort  of 
multiplication ;  for  they  ought  to  impress  the  results  upon  the 
memory,  if  they  wish  subsequently  to  obtain  easily  the  product 
of  numbers  expressed  by  several  figures.  Nevertheless,  for  those 
who  are  suflBciently  exercised,  all  that  is  necessary  is  to  give  a 
table  called  the  multiplication  tabUj  or  tahle  of  Pythagoras,  from 
the  name  of  its  inventor,  or  at  least  from  him  who  first  brought 
it  into  public  use. 


1 

1   2 

•^ 

1  4 

5 

1   « 

7 

1   « 

1  9 

2  , 

1  4 

6 

1  8 

10 

|12 

14 

116 

|18 

3 

1  6 

9 

|12 

15 

|18 

21 

|24 

|27 

4 

1  8 

12 

|16 

20 

|24 

28 

|82 

36 

5 

|10 

15 

|20 

25 

|30 

35 

140 

|45 

6 

|12 

18 

|24 

30 

|36 

42 

|48 

|54 

7 

|14 

21 

|28 

35 

42 

49 

1  56 

1  63 

8 

|16 

24 

132 

40 

48 

50 

|64 

|72 

9 

|18 

27 

36 

45 

54 

63 

|72 

81 

The  first  horizontal  row  of  this  table  is  formed  by  adding  1  to 
itself  up  to  9 ;  the  second,  by  adding  2  to  itself;  the  third,  by 
adding  3 ;  and  so  on  for  the  rest.  We  remark,  moreover,  that 
the  same  arrangement  is  made  in  the  vertical  columns.  Each 
vertical  column,  taken  in  order,  is  composed  of  the  same  num- 
bers as  each  horizontal  row.     Thus,  the  sixth  horizontal  row  is 


28  MULTIPLICATION. 

composed  of  6,  12,  18... 54,  and  the  sixth  vertical  column  is 
composed  of  the  same  numbers,  6,  12,  18... 54. 

That  being  established,  when  we  wish  to  obtain  the  product 
of  two  numbers  from  this  table,  we  seek  the  multiplicand  in  the 
first  horizontal  row,  and  go  down  from  this  number  vertically, 
until  we  arrive  at  that  one  which  is  opposite  to  the  multiplier, 
which  we  find  in  the  first  vertical  column.  This  number,  con- 
tained in  the  little  square,  is  the  product.  For  example,  in  order 
to  find  the  product  of  8  by  5,  we  descend  from  8,  taken  in  the 
first  horizontal  row  opposite  to  5  in  the  first  vertical  column,  and 
the  number  40  in  the  little  square  is  the  required  product. 

19.  Suppose,  now,  that  the  multiplicand  consists  of  several 
figures,  and  the  multiplier  of  a  single  figure. 

OPIRATION. 

8459  Let  it  be  proposed  to  multiply  8459  by  7.     We 

8459  could  (17)  obtain  the  result  by  writing  one  under  an- 
8459  other  seven  numbers  equal  to  8459,  and  adding  suc- 
8459  cessively  the  simple  units,  the  tens,  hundreds,  &c., 
8459  together.  We  would  thus  find  59213  for  a  result.  But 
8459  it  is  evident  that  this  is  nothing  more  than  taking 
8459  7  times  the  9  units  of  the  multiplicand,  7  times  the  5 
59213  tens,  &c.,  and  then  to  take  the  sum  of  all  the  pro- 
ducts. 
8459  Thus,  after  having  placed  the  multiplier,  7,  under 

7  the  multiplicand,  we  say,  7  times  9  make  63,  (see 
59213  table  of  multiplication),  or  6  tens  and  3  units;  we 
place  the  3  under  the  units,  and  reserve  the  6  tens  in 
order  to  add  them  to  the  product  of  the  tens  of  the  multiplicand 
by  7.  We  thus  say,  7  times  5  make  35,  and  6  make  41  tens,  or 
4  hundreds  and  1  ten ;  we  place  1  in  the  column  of  tens,  and 
reserve  the  4  hundreds ;  7  times  4  make  28,  and  4  make  32 
hundreds,  or  3  thousands  and  2  hundreds;  we  place  2  in  the 
column  of  hundreds,  and  retain  the  3 ;  lastly,  7  times  8  make 
56,  and  3  make  59 ;  we  write  down  the  9,  and  carry  the  5  one 
place  to  the  left,  because  there  are  no  more  figures  in  the  multi- 


MULTIPLICATION.  29 

plicand  to  be  multiplied.     We  find  thus,  59213  for  the  required 
product. 

Whence  we  see  that,  in  order  to  multiply  one  number  of 
several  Jigures  hy  another  of  a  single  figure,  we  must  multiply 
successively  the  units,  tens,  hundreds,  dsc,  of  the  midtiplicand  hy 
the  multiplier,  and  write  these  different  palatial  products  in  the 
columns  to  which  they  belong,  taking  care  at  each  partial  multi- 
plication, to  reserve  the  tens  in  order  to  add  them  to  the  tens,  the 
hundreds  in  order  to  add  them  to  the  hundreds,  &c. 

OPERATION. 

Let  it  be  proposed  as  a  second  example  to  multiply         37008 

37008  by  9.     We  say,  first,  9  times  8  make  72 ;  we       9 

write  2  in  the  column  of  units,  and  reserve  the  7.  333072 
Then  9  times  0  give  0 ;  but  we  have  reserved  7  from 
the  preceding  operation,  so  we  write  these  7  tens  in  the  column 
of  tens ;  9  times  0  make  0 ;  we  write  0  in  the  rank  of  hundreds, 
since  there  are  none,  and  since  it  is  necessary  to  preserve  the 
place  of  hundreds ;  then  9  times  7  make  63 ;  we  set  down  3  and 
reserve  6 ;  lastly,  9  times  3  make  27,  and  6  make  33 ;  we  set 
down  3,  and  advance  3  one  place  to  the  left.  Thus,  the  required 
product  is  333072. 

20.  Before  passing  to  the  case  in  which  the  multiplier  is  com- 
posed of  two  or  more  figures,  we  will  explain  the  method  of 
rendering  a  number  10,  100,  1000  times  greater,  or  of  multiply- 
ing it  by  10,  100,  1000. 

It  results  from  the  fundamental  principle  of  numeration  (5), 
that  if  we  place  a  0  to  the  right  of  a  number  already  written, 
each  one  of  the  significant  figures  of  the  number  being  thus 
advanced  one  step  towards  the  left,  expresses  units  ten  times 
greater  than  before.  In  the  same  manner,  by  placing  two  O's  to 
the  right,  we  render  it  100  times  as  great,  because  each  signifi- 
cant figure  expresses  units  100  times  as  great. 

Then,  in  order  to  multiply  any  entire  number  whatever  by  10, 
100,  1000,  &c.,  it  sufires  to  annex  1,  2,  S,... zeros. 
3  *  " 


•SO  MULTIPLICATION. 

Thus,  the  products  of  439  by  10,  100,  1000,  10,000,  &c.,  are 
4390,  48,900,  439,000. 

21.  Let  us  consider  now  the  case  in  which  the  multiplicand 
and  multiplier  are  composed  of  several  figures. 

OPERATION. 

We  propose  to  multiply  87468 

By :.....         5847 

We  commeDce  by  placing  the  multiplier  under  612276 

the  multiplicand,  so  that  the  units  of  the  same  3498720 

order  fall  in  the  same  column.  This  being  ar-  69974400 
ranged,  we  observe  that,  to  multiply  87468  by  437340000 
5847,  is  to  take  the  multiplicand,  7  times,  40  511425396 
times,  800  times,  and  5000  times;  then  to  add 
together  these  partial  products.  We  can  first  find,  by  the  rule 
of  (19)  the  product  of  87468  by  7,  which  gives  612276.  But 
how  obtain  that  of  87468  by  40  ?  Let  us  conceive,  for  an  instant, 
that  we  have  written,  one  under  another,  40  numbers  equal  to 
87468,  and  that  we  make  the  addition  of  these  numbers ;  we 
will  thus  have  the  required  product.  But  it  is  evident  that  these 
40  numbers  form  ten  divisions,  each  division  containing  4  times 
87468.  We  form  this  product  by  rule  (19),  and  find  it  to  be 
349872.  Multiplying  this  product  by  10,  which  (20)  is  efi'ected 
by  annexing  a  0,  we  obtain  3498720  for  the  product  of  87468 
by  40. 

We  see,  then,  that  this  second  operation  reduces  itself  to  mul- 
tiplying the  multiplicand  by  the  figure  4,  considered  as  express- 
ing simple  units,  in  writing  a  0  to  the  right  of  the  product,  and 
in  placing  the  result  as  we  see  above,  below  the  first  partial  pro- 
duct. In  like  manner,  in  order  to  perform  the  multiplication  of 
87468  by  800,  it  suffices  to  multiply  87468  by  8,  which  gives 
699744;  then  annex  two  O's  to  the  right  of  this  product;  we 
thus  have  a  third  partial  product,  69974400,  which  we  place 
below  the  two  preceding  products.  For  800  numbers,  equal  to 
87468,  and,  placed  one  under  another,  form  evidently  100  divi- 
sions of  8  numbers,  each  equal  to  87468,  or  100  numbers,  equal 
to  the  product  of  87468  by  8 ;  that  is  to  say,  6997400      We 


MULTIPLICATIOx\.  31 

could  prove  by  a  similar  course  of  reasoning,  that,  in  order  to 
multiply  by  5000,  it  suffices  to  multiply  by  5,  to  annex  three 
zeros  to  the  product,  and  write  the  result,  437340000,  thus  ob- 
tained, below  the  three  first  products.  Performing  now  the  addi- 
tion of  these  four  partial  products,  we  find  at  last  the  total  pro- 
duct, 511425396. 

N.  B.  —  In  practice,  we  dispense  ordinarily  with  adding  the 
zeros  to  the  right  of  the  partial  products,  found  by  multiplying 

by  the  figures  in  the  tens,  hundreds, places;  but  we  write 

each  partial  product  below  the  preceding  product,  advancing  it 
one  place  to  the  right  with  reference  to  this  product ',  that  is  to 
say,  we  make  its  last  figure  occupy  the  same  column  which  the 
figure  by  which  we  multiply,  occupies. 

General  Rule.  —  In  order  to  multiply  a  number  of  several 
figures  by  a  number  of  several  figures — Multiply  first  the  multi- 
plicand hy  the  units  figure  of  the  inultiph'er,  after  the  rule  of 
(19) )  multiply  in  the  same  manner  the  whole  multiplicand,  suc- 
cessively by  the  tens  figure,  by  that  of  hundreds,  &c.,  considered 
as  simple  units,  and  write  the  partial  products  one  under  the 
other,  so  that  each  one  is  advanced  one  column  to  the  left,  with 
reference  to  the  preceding ;  then  add  these  products;  the  respJt 
will  be  the  total  required  produx^t. 

22.  Often  some  of  the  figures  of  the  multiplier  are  zeros,  and 
then  it  is  necessary  to  make  some  modifications  in  the  arrange- 
ment of  the  partial  products. 

OPERATION. 

Multiply 870497 

By  500407 

We  multiply,  first,  th«  whole  multiplicand  by  6093479 

7,  which  gives  for  a  product  6093479.     Now,  3481988 

as  there  are  no  tens  in  the  multiplier,  we  pass     43524S5 
to  the  multiplication  by  4,  the  hundreds  figure,     435002792279 
which  gives  the  product,  3481988  ;  and,  since  it 
is  necessary  to  make  it  express  hundreds,  we  place  it  under  the 
first  product,  advancing  it  two  columns  to  the  left.    In  like  man- 


32 


MULTIPLICATION. 


ner,  as  there  are  no  thousands,  nor  tens  of  thousands  in  tlie  mul- 
tiplier, we  pass  to  the  multiplication  by  5,  the  figure  in  the  place 
of  hundreds  of  thousands,  and  write  the  product,  4352485,  under 
the  preceding,  advancing  it  three  places  to  the  left,  with  reference 
to  that  one. 

In  general,  when  there  are  one  or  more  zer&&  between  two 
significant  figures  of  the  multiplier,  we  advance  the  product 
corresponding  to  the  significant  figure,  which  is  to  the  left  of  these 
zeros,  one  more  column  to  the  left  than  there  are  zer'os  between  the 
figures. 

In  fine,  in  order  to  avoid  all  error  on  this  subject,  we  must 
take  care  at  each  operation  that  the  last  figure  of  each  partial 
product  falls  in  the  column  of  units  of  the. same  order  as  that  of 
the  figure  by  which  we  multiply. 

23.  If  one  of  the  two  factors  of  the  multiplication,  or  both, 
are  terminated  by  zeros,  we  abridge  the  operation  by  multiplying 
them  as  if  the  zeros  were  not  there ;  but  we  place  them  at  the 
end  of  the  product. 

EXAMPLE. 

OPEBATIOir. 

Multiply 47000 

By .' 2900 

After  having  multiplied  47  by  29,  according  to  423 

the  known  method,  we  annex  5  zeros  to  the  right  94 


of  the  product,  and  thus  obtain  136300000  for  136300000 
the  required  product.  For,  if  we  had  at  first  to 
multiply  47000  only  by  29,  it  would  be  necessary  to  make  the 
product  express  thousands  (i.  e.)  units  of  the  same  species  as  the 
multiplicand;  thus  we  ought  to  add  3  zeros.  But  to  multiply  a 
number  by  2900,  is  (21)  to  take  100  times  the  product  by  29 ; 
then  we  must  add  two  new  zeros.  The  same  reasoning  applies  to 
all  similar  cases. 

24.  But  little  reflection  on  the  method  of  multiplication  will 
convince  us  of  the  necessity  of  commencing  the  operation  on  the 


(   VNIVrR 

\HtLlT[PX.ICATION.  33 

right,  at  hast  in  the  partial  muhipllcaf  ion  hy  each  one  of  the  figures 
of  the  multiplier^  because  of  the  reservations  of  figures  which 
we  frequently  make  in  multiplying  each  figure  of  the  multipli- 
cand by  each  figure  of  the  multiplier. 

But  nothing  prevents  us  from  inverting  the  order  of  the  partial 
multiplications  by  the  different  figures  of  the  multiplier,  as  we 
can  see  in  the  following  example. 

"We  have  here  commenced  the  multiplication  with       operation. 
the  hundreds  figure  of  the  multiplier  3   but  in  the         5704 
following  operation  we  have  taken  care  to  advance  487 

the  product  one  column  to  the  right.     In  the  same       22816 
manner  the  third  product  is  advanced  one  place  to         45632 
the  right  with  reference  to  the  preceding.     Usage  39928 

alone  requires  us  to  form  the  products  from  right  to  2777848 
left;  it  is  also  the  more  natural  and  convenient  method. 

25.  We  will  close  the  subject  of  multiplication  by  the  ex- 
planation of  several  properties,  of  which  we  will  often  have  to 
make  use.    • 

1st.  Let  it  he  required  to  multiply, '^^b  hy  12,  equal  to  8  mul- 
tiplied hy  9.  We  say,  that  to  multiply  345  by  72,  is  to  multiply 
345  by  9,  and  the  result  by  8. 

In  order  to  establish  this  proposition  without  performing  i\\Q 
operations,  we  must  employ  a  mode  of  reasoning  analogous  to 
that  employed  in  (21).  To  multiply  345  by  72,  is  to  sum  up  72 
numbers  equal  to  345.  But  these  72  numbers,  written  one  under 
another,  form  evidently  8  divisions  of  9  numbers,  equal  to  345 ; 
then,  after  having  multiplied  345  by  9,  we  must  take  this  product 
8  times.  Thus,  to  multiply  345  by  the  product  72  of  the  two 
factors,  9  and  8,  is  to  multiply  345  by  9,  and  the  new  result  by  8. 
As  9  is  itself  equal  to  the  product  of  3  and  3,  we  can  say,  that 
to  multiply  345  by  72,  is  to  multiply  345  by  3,  the  result  obtained 
by  3,  and  finally  the  new  result  by  8.  As  we  can  apply  this  rea- 
soning to  other  numbers,  this  general  proposition  results  from  it : 
to  multiply  a  numher  hy  a  product  of  tico  or  more  numhers 
already  formed,  amounts  to  the  same  tltivg  as  multiplying  the 
numher  hy  each  one  of  the  factors  successively. 


34  .  Divisiox. 

26.  —  2d.  In  a  multiplication  of  two  factors,  we  can  talce  in- 
differently the  first  number  for  multiplicand,  the  second  for  mul- 
tiplier, or  reciprocally  In  other  terms  —  the  product  of  two 
numbers  is  the  same  in  whatever  order  we  perform  the  operation. 

Thus,  the  product  of  459  by  237,  is  equal  to  the  product  of 
237  by  459. 

For,  let  us  conceive  unity  written     1,  1,  1,  T,  1, 

459  times  in  a  horizontal  line,  and  let     1,  1,  1,  1,  1, 

us  form  237  of  these  lines ;  it  is  clear     1,  1,  1,  1,  1, 

that  the  sum  of  the  units  contained  in     1,  1,  1,  1,  1, 

such  a  figure  is  equal  to  as  many  times 

the  459  units  of  a  horizontal  row  as 

there  are  units  in  a  vertical  column,  or 

in  237,  (i.  e.)  that  this  sum  is  equal  to  the  product  of  459  by  237. 
But  we  can  say  also,  that  this  sum  is  equal  to  as  many  times  the 
237  units  of  the  vertical  column  as  there  are  units  in  a  horizontal 
row,  or  in  459 ;  that  is  to  say,  is  equal  to  the  product  of  237  by 
459.  Then,  &c.  If  the  nature  of  a  question  con'ducts  to  the 
multiplication  of  76  by  5672,  according  to  the  proposition  which 
we  have  just  demonstrated,  we  would  prefer  to  take  the  product 
of  5672  by  76,  because  in  that  case  we  would  only  have  two 
partial  products  to  form,  while  in  the  other  operation  we  would 
have  to  form  four  of  them.  This  proposition  will  be  demonstrated 
for  any  number  whatever  of  factors. 

DIVISION. 

27.  To  divide  one  num.ber  by  another,  is  to  find  a  third  num- 
ber, which,  multiplied  by  the  second,  will  reproduce  the  first ;  or, 
in  other  terms,  being  given  the  product  and  one  of  the  factors,  to 
determine  the  other  factor.  As  in  the  multiplication  of  entire 
numbers,  the  product  is  composed  of  as  many  times  the  multipli- 
cand as  there  are  units  in  the  multiplier,  we  can  also  say,  that, 
to  divide  one  entire  number  by  another,  is  to  seek  how  many 
times  the  first  number,  considered  as  a  product,  contains  the  se- 


DIVISION.  35 

cond,  considered  as  multiplicand ;  the  number  of  times  is  then 
the  multiplier.  Finally,  we  can  also  say,  that,  to  divide  a  numher 
hi/  another,  is  to  divide  the  first  number  into  as  many  equal  parts 
as  there  are  units  in  the  second. 

These  last  two  points  of  view,  under  which  we  sometimes  con- 
sider division,  pertain  only  to  entire  numbers,  while  the  two  first 
pertain  to  all  possible  numbers,  whether  entire  or  fractional. 
Nevertheless,  the  names  given  to  the  terms  of  division  have  been 
drawn  from  these  last  two  points  of  view. 

Thus,  the  first  number  is  called  dividend,  the  second  is  called 
divisor,  and  the  third  quotient,  from  the  Latin  word  quoties ; 
because  it  expresses  how  many  times  the  dividend  contains  the 
divisor. 

It  results,  obviously,  from  the  first  two  definitions,  that  when 
we  have  obtained  the  quotient,  in  order  to  make  the  verification 
of  the  operation,  it  will  suffice  to  multiply  the  divisor  by  tha 
quotient ;  and,  if  the  operation  has  been  exact,  we  will  thus  re- 
produce the  dividend.* 

Reciprocally  in  multiplication,  the  product  may  be  considered 
as  the  dividend,  the  multiplicand  as  the  divisor,  and  the  multi- 
plier as  the  quotient ;  thus,  we  make  the  verification  of  multipli- 
cation by  dividing  the  product  by  one  of  the  factors ',  and  if  the 
operation  is  exact,  we  ought  to  reproduce  the  other  factor.  These 
ideas  being  established,  we  pass  to  the  explanation  of  the  method 
of  division. 

28.  In  the  same  manner  as  multiplication  can  be  effected  by 
the  addition  of  a  number  several  times  to  itself,  we  can  also  find 
the  quotient  of  a  division  by  a  series  of  subtractions. 

For,  let  it  be  required  to  divide  60  by  12.  As  many  times  as 
we  can  subtract  12  from  60,  so  many  times  is  12  contained  in 
60.  Thus,  the  quotient  is  equal  to  the  number  of  subtractions 
which  we  can  make  before  the  dividend  is  exhausted. 


30 


DIVISION. 


60        In  this  example,  as  we  are  obliged  to  make  5 

1^     subtractions,  it  follows  that  the  quotient  is  5.  But 

1st  rem.    48     this  manner  of  performing  the  division  would  be 

12     too  long  in  practice,  especially  if  the  dividend  was 

2d  rem.    36     very  great  in  comparison  with  the  divisor.     It  is 

12     in  the  art  of  abridging  the  operation  that  the  or- 

3c?  rem.    24     dinary  method  of  division  consists: 

12         29.  From  the  fact  that  we  know  by  heart  the  pro- 
4it7i  7'em.   12     ducts  of  two  numbers  of  a  single  figure,  we  can 
12     determine  easily  the  quotient  of  the  division  of  a 
bth  rem.     0     number  of  one  or  two  figures  by  a  number  of  a 
single  figure. 
For  example,  35  divided  by  7,  gives  for  a  quotient  5.     This 
we  know,  because  we  know  that  7  times  5  give  35.     We  say, 
also,  in  this  example,  that  the  7th  of  35  is  5,  because  7  times  5 
make  35.     Suppose,  again,  that  we  have  to  divide  68  by  9.     As 
7  times  9,  or  63,  and  8  times  9,  or  72,  comprise  68  between 
them,  it  follows  that  68,  divided  by  9,  gives  for  the  quotient,  7, 
with  a  remainder,  5 ;  or  the  9th  of  68  is  7,  .with  a  remainder,  5. 
In  like  manner,  47  contains  8,  5  times,  with  a  remainder  7 ; 
because  5  times  8  gives  40,  and  6  times  8  gives  48. 

We  will  see  farther  on  what  is  to  be  done  with  the  remainder, 
when  the  divisor  is  not  contained  exactly  in  the  dividend. 

30.  Let  us  consider  the  case  in  which  the  dividend  is  com- 
posed of  any  number  of  figures,  the  divisor  containing  but  a 
single  figure. 

Divide  6766453  by  8. 
6766453)8 


64 

845806 

36 

32 
46 

Proof 

by  multiplication. 
845806 

40 

8 

64 

6766448 

64 

6 

053 

6766458 

48 

5 

DIVISION.  37 

After  having  written  the  divisor  to  the  right  of  the  dividend, 
and  separated  them  by  a  vertical  line,  we  draw  below  the  divisor 
a  horizontal  line.  This  arranged,  We  see  at  once  that,  if  we  place 
(mentally)  to  the  right  of  the  divisor,  8,  five  zeros,  (i.  e. )  multi- 
ply it  by  10,000,  then  six  zeros,  or  multiply  it  by  100,000,  the 
two  products,  80,000  and  800,000,  are  the  one  smaller,  the  other 
greater  than  the  dividend.  Whence  we  conclude  that  the  quo- 
tient demanded  is  comprised  between  10,000  and  100,000 ;  that 
is  to  say,  is  composed  of  six  figures,  and  that  thus  the  highest 
units  of  the  quotient  are  hundreds  of  thousands,  of  which  we 
must  find  the  figure. 

Now,  as  the  product  of  the  divisor  by  the  figure  sought  cannot 
give  units  of  a  lower  order  than  hundreds  of  thousands,  it  fol- 
lows, that  this  product  is  contained  wholly  in  the  67  hundreds 
of  thousands  of  the  dividend;  and  if  we  divide  67  by  8,  which 
gives  the  quotient  8  for  64,  and  the  remainder  3,  we  can  affirm 
that  the  figure  of  hundreds  of  thousands  in  the  quotient  is  8. 
In  fact,  800,000  times  8  gives  6,400,000,  a  number  which  can 
be  subtracted  from  the  dividend,  6766453 ;  while  900,000  times 
8,  or  7,200,000  cannot  be  so  subtracted.  The  figure  8  being  thus 
determined,  we  place  it  under  the  divisor;  then  we  subtract  the 
product  8  by  8,  or  64  from  67,  and  conceive  the  remaining 
figures  of  the  dividend  to  be  written  to  the  right  of  the  remain- 
der 3,  which  gives  366453  for  the  total  remainder  of  this  first 
operation.  (This  first  operation  is  evidently  nothing  more  than 
subtracting  from  the  dividend  800,000  times  the  divisor,  or  is 
equivalent  to  800,000  successive  subtractions  of  the  divisor  8.) 
It  would  seem  necessary  to  write  on  the  right  of  the  quotient 
already  obtained,  five  zeros,  in  order  to  give  it  its  true  value ; 
but  we  avoid  this  by  the  arrangement  which  we  will  make  of  the 
following  figures  of  the  quotient. 

We  must  now  determine  the  figure  of  tens  of  thousands  of  the 
quotient.  Since  the  product  of  the  divisor  by  this  figure  cannot 
give  units  of  an  order  inferior  to  tens  of  thousands,  it  is  con- 
tained wholly  in  the  36  tens  of  thousands  of  the  remaining  divi- 
dend. It  suffices  then  to  bring  down  to  the  side  of  the  remain- 
4 


38  DIVISION. 

der,  3,  the  following  figure,  6,  of  the  dividend ;  then  to  divide  36 
bj  8,  which  gives  the  quotient,  4,  for  32,  and  the  remainder,  4. 
We  write  this  quotient,  which  expresses  necessarily  the  tens  of 
thousands  of  the  whole  quotient,  on  the  right  of  the  first  quo- 
tient, 8 ;  then,  after  having  subtracted  4  times  8,  or  32  from  36, 
we  bring  down  to  the  right  of  the  4,  the  next  figure  of  the  divi- 
dend, which  gives  64.  (This  new  operation,  which  amounts  to 
subtracting  40,000  times  8,  or  320,000  from  366,453,  is  equiva- 
lent to  40,000  new  successive  subtractions  of  the  divisor,  8.) 

In  order  to  obtain  the  ones  of  thousands  of  the  whole  quo- 
tient, we  divide  46  by  8 ;  the  quotient  is  5  for  40,  and  the  re- 
mainder, 6.  We  write  this  new  quotient,  5,  to  the  right  of  the 
first  two ;  then,  after  having  subtracted  5  times  8,  or  40,  from  46, 
we  bring  down  to  the  right  of  the  remainder,  6,  the  next  figure, 

4,  of  the  dividend,  which  gives  64.  (This  third  operation  is 
equivalent  to  5000  successive  subtractions  of  the  divisor,  8.) 

In  order  to  obtain  the  figure  of  hundreds  of  the  total  quotient, 
we  divide  64  by  8,  which  gives  8,  and  0,  for  remainder;  we  write 
the  new  quotient  to  the  right  of  the  three  first ;  then,  after  having 
subtracted  8  times  8,  or  64  from  64,  we  bring  down  to  the  right 
of  the  remainder,  0,  the  next  figure  of  the  dividend,  which  gives 

05,  or  simply  5. 

Here  a  particular  case  presents  itself;  as  the  new  partial  divi- 
dend, 05  or  5,  which  is  to  give  the  tens  of  the  quotient,  is  less 
than  the  divisor,  8,  we  must  conclude  that  the  total  quotient  has 
no  tens,  (and  in  fact  the  remaining  dividend  is  53,  a  number  less 
than  10  times  8,  or  80.) 

We  place,  then,  a  0  in  the  quotient  to  the  right  of  the  four 
figures  already  obtained,  in  order  to  replace  the  tens  which  are 
wanting,  and  preserve  the  relative  value  of  the  preceding  and 
following  figures ;  we  then  bring  down  to  the  right  of  the  re- 
mainder, 5,  the  next  and  last  figure  of  the  dividend,  and  con- 
tinue the  operation.  The  quotient  of  53  divided  by  8,  being  6 
for  48,  we  write  this  figure  to  the  right  of  the  first  five  quotients 
already  found;  we  then  subtract  48  from  53,  which  gives  at  last 
5  for  the  remainder  of  the  entire  operation ;  and  the  required 


DIVISION.  39 

quotient  is  845806,  which  we  can  easily  verify  by  multiplying  8 
by  845806,  or  rather  848806  by  8,  and  adding  the  remainder,  5, 
to  the  product  thus  obtained.  (All  the  operations  which  have 
been  performed  in  eifecting  this  division  are  equivalent,  evidently, 
to  800,000,  then  40,000  subtractions,  then  5000,  then  800,  then 
6,  or  845806  successive  subtractions,  in  which  the  divisor,  8,  is 
constantly  the  number  to  be  subtracted.) 

31.  We  will  not  establish  for  the  case  of  division  which  we 
have  just  discussed,  a  general  rule  founded  on  the  preceding 
reasoning,  because  there  exists  (for  this  case  only)  a  practical 
method,  more  convenient  and  more  simple  in  reference  to  the 
arrangement  of  the  calculations.  Let  us  take  again  the  above 
example : 

6766453  to  be  divided  by  8. 
Quotient,  845806;  remainder,  5. 

We  know  already  (No.  27)  that  to  divide  a  number  by  8,  or  to 
seek  how  many  times  8  is  contained  in  this  number,  amounts  to 
dividing  the  number  into  8  equal  parts,  or  taking  the  eighth  of 
it.  This  being  fixed,  taking  the  two  first  figures  to  the  left  of 
the  dividend,  67,  we  say,  the  eighth  of  67  is  8,  with  the  re- 
mainder, 3.  We  write  the  quotient,  8,  under  the  figure^  7,  of 
the  dividend;  then  we  place,  mentally,  the  remainder,  3,  ex- 
pressing 3  hundreds  of  thousands,  or  30  tens  of  thousands,  to 
the  left  of  the  figure,  6,  of  the  dividend,  which  expresses  also 
tens  of  thousands ;  we  say,  as  before,  the  eighth  of  36  is  4,  with 
remainder  4.  We  write  the  second  quotient,  4,  to  the  right  of 
the  first;  placing  again,  mentally,  the  remainder,  4,  expressing 
4  tens  of  thousands,  or  40  thousands,  to  the  left  of  the  thousands 
figure,  6,  of  the  dividend ;  we  say,  again,  the  eighth  of  46  is  5, 
with  the  remainder,  6 ;  we  write  the  third  quotient,  5,  to  the 
right  of  the  preceding;  continuing  in  the  same  manner,  we  say, 
again,  the  eighth  of  64  is  8,  with  the  remainder,  0,  and  we  write 
the  fourth  quotient,  8,  to  the  right  of  the  third.  The  eighth  of 
05,  or  5,  is  0,  with  the  remainder,  5 ;  we  write  this  fifth  quotient 
to  the  right  of  the  fourth.     Finally,  the  eighth  of  53  is  6,  with 


40  DIVISION. 

the  remainder,  5 ;  we  write  to  the  right  of  the  fifth  quotient  the 
sixth  and  last  partial  quotient,  which  thus  falls  beneath  the  units 
figure  of  the  dividend^  and  we  have  for  the  result  the  quotient, 
845806,  with  the  remainder,  5. 

Second  example : 

8230200409  to  be  divided  by  6. 
Quotient,  1371700068;  remainder,  1. 

Here,  the  first  figure  on  the  left  of  the  dividend  being  greater 
than  the  divisor,  we  see  that  the  quotient  ought  to  have  units  of 
the  same  order  as  those  of  the  figure  8 ;  and  we  say,  the  sixth  of 
8  is  1,  which  we  write  under  the  figure  8,  with  the  remainder, 
2 ;  then  the  sixth  of  22  is  3,  which  we  place  to  the  right  of  the 
figure  1,  with  the  remainder,  4. 

The  6th  of  43  is  7,  with  the  remainder,  1. 

The  6th  of  10  is  1,  with  the  remainder,  1. 

The  6th  of  42  is  7,  with  the  remainder,  0. 

The  6th  of  0  is  0,  with  the  remainder,  0. 

The  6th  of  0  is  0,  with  the  remainder,  0. 

The  6th  of  4  is  0,  with  the  remainder,  4. 

The  6th  of  40  is  6,  with  the  remainder,  4. 

Finally,  the  6th  of  49  is  8,  with  the  remainder,  1. 

The  required  quotient  is  then  1371700068,  with  the  remain- 
der, 1. 

It  is  very  important  to  understand  thoroughly  this  method, 
because  it  finds  its  application  in  the  case  of  division,  which  is 
yet  to  be  discussed. 

We  will  observe,  moreover,  that  when  we  know  by  heart  the 
multiplication  table  as  far  as  the  number  12,  we  can  obtain  very 
easily,  by  the  same  method,  the  10th,  11th,  and  12th,  of  any 
number  whatever. 

EXAMPLES. 

1st.  897614708497,  to  be  divided  by  12. 
Quotient,  74801225708,  remainder,  1. 
(The  12th  of  89  is  7,  with  remainder,  5 ;  the  12th  of  57  is  4, 
with  remainder,  9;  the  12th  of  96  is  8,  remainder,  0;  &c.,  &c.) 


DIVISION.  41 

.2d.  23054273896,  to  be  divided  by  11. 
Quotient,  2095843081 ;  remainder,  5. 
(The  11th  of  23  is  2,  with  remainder,  1 ;  the  11th  of  10  is  0, 
with  remainder,   10;   the  11th  of  105  is  9,  with  remainder, 
6;  &c.,  &c.) 

As  to  the  division  by  10,  instead  of  applying  the  method,  it  is 
simpler  to  separate  in  thought  the  last  figure  to  the  right  of  the 
dividend.  The  part  to  the  left  expresses  the  quotient,  and  this 
last  figure  separated  (which  can  be  0),  is  the  remainder  of  the 
division.  This  is  an  evident  consequence  of  the  system  of  nu- 
meration. 

Thus,  the  10th  of  2710548  is  271054,  and  the  remainder,  8 ; 
the  10th  of  863005704  is  86300507,  and  remainder,  4;  the  10th 
of  3805670  is  exactly  380567 ;  results  which  can  be  found  also 
by  the  application  of  the  method  above. 

32.  Let  us  pass  to  the  case  in  which  the  given  numbers  being 
both  composed  of  several  figures,  the  quotient  is  to  have  one  only. 

This  case  deserves,  of  itself,  particular  attention ;  and  it  will 
serve  us,  besides,  as  a  basis  for  the  development  of  the  general 
case. 

Let  it  be  given  to  divide  730465  by  87467. 
87467 


730465 
699736 

"^729 

We  remark,  first,  that  the  product  of  the  divisor  by  10,  or 
874670,  is  greater  than  the  dividend ;  thus,  the  quotient  sought 
is  less  than  10,  and  can  have  only  one  figure. 

In  the  second  place,  the  product  of  8  tens  of  thousands  of  the 
divisor  by  the  figure  sought,  as  it  cannot  give  units  of  an  order 
inferior  to  tens  of  thousands,  must  be  found  wholly  in  the  73 
tens  of  thousands  of  the  dividend )  whence  it  follows,  that  the 
figure  sought  cannot  exceed  the  quotient  of  the  division  of  73 
by  8.  We  are  then  conducted  to  the  division  of  the  part,  73, 
on  the  left  of  the  dividend,  by  the  first  figure,  8,  of  the  divisor, 
4* 


42  DIVISION. 

wHcli  gives  the  quotient,  9.  But  9  is  evidently  too  large;  for, 
in  the  multiplication  of  the  whole  divisor  by  this  figure,  we  find, 
in  multiplying  the  thousands  figure,  7,  of  the  divisor,  by  9,  63 
units  of  this  order,  and,  consequently,  6  tens  of  thousands,  to  be 
added  to  the  72  tens  of  thousands,  product  of  the  first  figure,  8, 
of  the  divisor,  by  the  same  figure,  9 ;  which  would  give  78  tens 
of  thousands,  a  number  greater  than  the  dividend 

It  is  not  necessary,  then,  to  try  any  figure  higher  than  8,  as 
figure  of  the  quotient  required.  Effecting  the  multiplication  of 
87467,  by  8  (which  we  have  placed  under  the  divisor),  we  ob- 
tain a  product  of  699736,  less  than  the  dividend ;  which  proves 
that  the  quotient,  8,  is  correct.  On  subtracting  this  product 
from  the  dividend,  as  the  operation  shows,  we  find  for  remainder, 
80729. 

Again,  divide  974065  by  189768. 

974065  1 189768 

948840  I  5 


25225 

As  the  dividend  and  the  divisor  are  composed  of  the  same 
number  of  figures,  it  is  clear  that  the  quotient  ought  to  have 
only  one  figure;  and  in  order  to  find  it,  we  divide,  first,  the  first 
figure  on  the  left  of  the  dividend,  9,  by  the  first  figure,  1,  on 
the  left  of  the  divisor.  The  quotient  is  9 ;  but  this  figure,  and 
the  next  lower,  8,  7,  6,  are  too  large,  if  we  consider  the  two  first 
figures,  18,  on  the  left  of  the  divisor;  for  the  products  of  18, 
by  9,  8,  7,  6,  being  162,  144,  126,  and  108,  all  surpass  the  97 
tens  of  thousands  of  the  dividend.  This  leads  us  to  try  the 
figure  5. 

On  multiplying  the  divisor  by  5,  we  have  the  product,  948840, 
which,  subtracted  from  the  dividend,  gives  for  remainder,  25225, 
a  number  smaller  than  the  divisor ;  which  proves  that  the  quo- 
tient, 5,  is  not  too  small. 

33.  In  the  two  preceding  examples,  we  have  been  able  to  de- 
termine pretty  easily  what  was  the  true  figure  of  the  quotient ; 
but  as  this  is  not  always  the  case,  it  is  important  to  have  a  me- 


DIVISION.  43 

thod  of  ascertaining,  without  effecting  tlie  product  of  tlie  divisor 
by  the  quotient,  whether  the  trial  figure  is  the  true  one.  We 
will  now  develop  this  method. 

Particular  method  of  trial. 
Given,  556428,  to  be  divided  by  69784. 
556428  I  69784 
488488  I  7 


•  67940 

The  division  of  55  (the  two  first  figures  on  the  left  of  the 
dividend),  by  6  (the  first  figure  on  the  left  of  the  divisor),  gives 
9  for  quotient,  with  the  remainder,  1. 

In  order  that  9  may  not  be  larger  than  the  quotient  sought,  9 
times  the  divisor  must  be  less,  or,  at  most,  equal  to  the  dividend; 
or,  which  is  the  same  thing,  the  9th  of  the  dividend  must  be 
greater,  or  at  least  equal  to  the  divisor. 

We  then  commence  to  take  the  9th  of  556428,  after  the  me- 
thod of  (31).  We  find  for  the  two  first  figures  on  the  left,  61 
tens  of  thousands,  a  number  less  than  69  tens  of  thousands  of 
the  divisor,  which  shows  that  the  9th  of  the  dividend  is  less  than 
the  divisor;  9  ought  then  to  be  rejected.  We  next  try  8.  We 
find  for  the  three  first  figures  of  the  8th  of  the  dividend,  695 
hundreds  less  than  the  697  hundreds  of  the  divisor;  then  8  is 
too  large.  We  now  try  7.  The  first  figure  of  the  7th  of  the 
dividend  is  7,  greater  than  6,  the  first  figure  of  the  divisor. 
Whence  it  follows,  that  the  7th  of  the  dividend  is  greater  than 
the  divisor ;  or,  in  other  terms,  that  the  product  of  the  divisor 
by  7,  is  less  than  the  dividend.  Thus,  the  figure  7  is  the  true 
one.  Multiplying  the  divisor  by  7,  and  writing  the  product, 
488488,  below  the  dividend,  then  effecting  the  subtraction,  we 
obtain  the  remainder,  67940,  a  number  smaller  than  the  divisor. 

Another  Example. 
Given,  to  divide  1148367  by  169987. 


1148367 
1019922 

"128445 


169987 
6 


44  DIVISION. 

The  division  of  11  by  1,  would  give  11  for  a  quotient;  but 
the  required  quotient  cannot  be  greater  than  9,  siq.ee  the  <iivisor, 
multiplied  by  10,  would  be  a  larger  number  than  the  dividend. 

Let  us  try  9 ;  the  9th  of  the  dividend  is  12  ....  smaller  than 
the  divisor,  16 We  therefore  reject  the  9. 

The  8th  of  the  dividend  is  14  ...  .  less  than  16  ...  .  We 
therefore  reject  8. 

The  7th  of  the  dividend  is  164  ...  .  less  than  169  ....  of 
the  divisor.     We  then  reject  7. 

The  6th  of  the  dividend  is  19  ...  .  greater  than  the  divisor, 
16;  thus,  the  figure  6  is  the  true  one.  Multiplying  the  divisor 
by  6,  and  subtracting  the  product,  1019922,  from  the  dividend, 
we  obtain  the  remainder,  128445,  smaller  than  the  divisor. 

The  course  to  be  pursued  in  this  method  of  trial  is  evident 
from  the  exposition  of  the  last  example.  We  stop  as  soon  as  we 
obtain  a  figure  greater  or  less  than  the  corresponding  figure  of 
the  divisor.  If  it  is  greater,  we  can  affirm  that  the  trial  figure 
is  the  true  one ;  if  less,  we  know  that  the  trial  figure  must  be 
diminished. 

We  add,  that  all  these  trials  can  be  made  mentally,  without 
writing  anything. 

It  could  happen  (but  rarely),  that  we  reproduced  thus  suc- 
cessively all  the  figures  of  the  divisor,  in  arriving  at  a  remainder 
necessarily  less  than  the  trial  figure,  and  possibly  0.  We  would 
then  have  not  only  the  required  quotient,  but  also  the  remainder 
of  the  proposed  division,  which  would  be  nothing  else  than  this 
final  remainder. 

We  recommend  especially  the  exercise  of  this  method  of  trial, 
as  a  means  of  avoiding  all  difficulty  in  any  case  whatever. 

34.  /Second  Remark.  —  We  have  seen  in  all  which  precedes, 
that  after  having  determined  a  figure  of  the  quotient,  we  proceed 
to  multiply  the  divisor  by  this  figure,  to  write  the  product  below 
the  dividend,  and  to  effect  the  subtraction,  placing  the  remainder 
under  this  product.  But  we  can  employ  a  method  of  abbrevia- 
tion which  we  will  now  explain. 


DIVISION.  45 


Let  us  take  the  first  example  discussed  in  (32.) 


730465 


307:^9 


87467 


8 


This  method  of  abbreviation  consists  in  forming  the  product 
of  the  divisor  by  the  figure  8  of  the  quotient,  mentally,  and  in 
writing  only  the  remainder  below  the  dividend.  To  accomplish 
this,  we  must  subtract,  successively,  from  the  units,  tens,  hun- 
dreds, &c.,  of  the  diyidend,  the  products  of  the  units  of  the 
same  order  of  the  divisor  by  the  quotient,  as  fast  as  we  form  them 
mentally.  Thus,  in  the  example  above,  we  have  at  first  to  sub-  • 
tract  from  the  5  units  of  the  dividend  the  product,  56,  of  the 
quotient,  8,  by  the  7  units  of  the  divisor;  but  as  this  subtraction 
is  impossible  (which  will  generally  be  the  case),  we  add  mentally 
6  tens  to  the  5  units,  with  the  express  reservation  of  adding,  by 
way  of  compensation,  these  6  tens  also  to  the  product  of  the 
quotient,  8,  by  the  tens  of  the  divisor  (to  be  subtracted  in  its 
turn) ;  we  form  thus  the  number  65,  from  which  we  subtract  56, 
which  gives  for  a  remainder,  9,  which  we  write  below  the  units 
of  the  dividend.  Passing  to  the  tens  figure  of  the  divisor,  6, 
we  say,  8  times  6  give  48  tens,  which,  augmented  by  6  tens 
(reserved  in  the  last  operation),  make  54  tens  to  be  subtracted 
from  the  6  tens  of  the  dividend.  In  order  to  perform  the  sub- 
traction, we  add  5  hundreds  (tens  of  tens),  to  these  6  tens, 
making  56  tens,  from  which  we  subtract  the  54  tens,  and  write 
the  remainder,  2,  under  the  tens  of  the  dividend.  Continuing 
thus,  we  say,  8  times  4  hundreds  make  32,  and  5  (which  were 
added  in  the  preceding  operation),  make  37.  This  cannot  be 
subtracted  from  4 ;  but  37  from  44  leave  7,  which  we  write  under 
the  hundreds  of  the  dividend. 

In  the  same  manner,  8  times  7  make  56,  and  4  (added  in  the 
preceding),  make  60 ;  60  from  60  leave  0,  which  we  write  under 
the  thousands  of  the  dividend.  Finally,  8  times  8  make  64,  and 
6  make  70 ;  70  from  73,  leave  3.  The  total  remainder  is  then 
30729. 

Let  us  take,  again,  the  second  example.     We  abridge  the  dis- 


46  DIVISION. 

cussion,  making  use  of  tlie  expressions  commonly  used  in  prac- 
tice (though  often  incorrect). 

974065  1 189768 


25225 1  5 

Having  found  5  the  true  figure  of  the  quotient,  we  say,  5  times 
8  make  40 ;  40  from  45,  leave  5,  and  carry  4  (understood,  in 
order  to  add  them  to  the  following  product)  ;  5  times  6  make  30, 
and  4  make  34 ;  34  from  36,  2  remain,  and  carry  3 ;  5  times  7 
make  35,  and  3  make  38 ;  38  from  40,  remain  2,  and  carry  4 ; 
5  times  9  are  45,  and  4  make  49 ;  49  from  54  leave  5,  and  carry 
5.  Finally,  5  times  18  make  90,  and  5  make  95 ;  95  from  97 
leave  2.     The  total  remainder  is,  25225. 

N.  B.  It  is  very  important,  at  each  partial  operation,  to  say, 
carry  such  a  figure,  in  order  not  to  forget  the  number  which,  by 
compensation,  is  to  be  added  to  the  following  product. 

35.  We  have  thus  very  fully  discussed  the  simple  cases  of  divi- 
sion, because,  having  once  mastered  these  thoroughly,  also  the  me- 
thods belonging  thereto,  of  abbreviation  and  trial,  the  pupil  will 
find  no  difficulty  in  comprehending  the  general  case  which  we  will 
now  discuss,  namely,  lohere  the  dividend,  the  divisor,  and  the 
quotient,  contain  any  number  of  figures. 

GENERAL   CASE   OP   DIVISION. 

Given,  to  divide  9176298  by  2678. 


9176298 

2678 

Verification  by  multiplication, 

11422 

3426 

2678 

'       7109 

3426 

17538 

16068 

1470 

5356 
10712 
8034 
1470 

9176298 


DIVISION.  47 

We  arrange  here  the  parts  of  tlie  division ;  the  quotient,  the 
successive  remainders,  according  to  the  indications  which  pre- 
cede; then  we  reason  as  in  (No.  30). 

If  we  place  mentally  three  zeros,  and  afterwards  four  zeros  to 
the  right  of  the  divisor,  we  obtain  two  products,  2678000,  and 
26780000,  the  one  less,  the  other  greater  than  the  dividend. 
Thus,  the  total  quotient  is  comprised  between  1000  and  10,000, 
or  must  be  composed  of  four  figures,  of  which  the  first  to  the  left 
expresses  thousands.  In  order  to  find  this  first  figure,  we  observe 
that  its  product  by  the  divisor,  inasmuch  as  it  is  thousands,  is  to 
be  found  wholly  in  the  part,  9176  thousands  of  the  dividend. 
We  are  then  led  to  divide  9176  (which  we  consider  as  a  first 
partial  dividend),  by  2678 ;  and  the  greatest  number  of  times 
that  the  second  number  is  contained  in  the  first,  represents  the 
thousands  figure  of  the  total  quotient.  Now,  the  true  quotient 
of  9176,  by  2678,  obtained  according  to  the  method  of  trial  in- 
dicated in  (33),  is  3.  We  write,  then,  3  below  the  divisor;  we 
next  subtract  from  the  dividend,  the  product  of  the  divisor,  by  3, 
either  by  placing  this  product  below  the  partial  dividend,  and 
subtracting,  or  (No.  34),  effecting  simultaneously  the  subtraction 
and  the  multiplication,  as  the  table  above  indicates.  (This  first 
operation  amounts,  evidently,  to  subtracting  3000  times  the 
divisor  from  the  dividend.) 

The  remainder  of  this  first  subtraction  being  1142,  if  we  write 
after  it  the  figures  of  the  dividend,  which  have  not  yet  been  used, 
there  would  result  a  new  dividend,  upon  which  we  could  operate 
as  upon  the  primitive  dividend ;  but,  as  we  have  now  to  deter- 
mine the  hundreds  figure  of  the  quotient,  and  as  the  product  of 
the  divisor  by  this  figure  cannot  give  units  of  a  lower  order  than 
hundreds,  it  must  be  contained  wholly  in  the  11422  hundreds 
of  the  remaining  dividend ;  so  we  bring  down  to  the  right  of  the 
remainder,  1142,  only  the  following  figure,  2,  of  the  dividend ; 
which  gives  a  second  partial  dividend,  11422,  upon  which  we 
operate  as  on  the  first. 

The  true  quotient  of  the  division  of  11422,  by  2678,  is  4, 
which  we  write  below  the  divisor,  and  to  the  right  of  the  first 


48  DIVISION. 

quotient  obtained.  We  then  subtract  from  the  second  partial 
dividend;  the  product  of  ^he  divisor,  by  the  new  quotient.  The 
remainder  of  this  subtraction  is  710.  We  bring  d6wn  to  its  right 
the  following  figure  of  the  dividend,  9,  which  gives  a  third  divi- 
dend, 7109,  which  is  to  furnish  the  tens  figure  of  the  total  quo- 
tient. 

Dividing  7109  by  2678,  we  have  for  a  true  quotient,  2,  which 
we  write  to  the  right  of  the  two  first  figures  of  the  quotient ; 
multiplying  the  divisor  by  2,  and  subtracting  the  product  from 
the  third  partial  dividend,  we  obtain  1753  for  a  remainder,  to 
the  right  of  which  we  bring  down  the  last  figure,  8,  of  the  divi- 
dend, which  gives  17538  for  a  fourth  partial  dividend.  Finally, 
the  true  quotient  of  17538  by  2678,  is  6.  We  multiply  the 
divisor  by  6,  and  subtract  the  product  from  the  fourth  partial 
dividend,  which  gives  a  remainder,  1470.  The  required  quotient 
is  then  3426,  with  the  remainder,  1470 ;  which  we  can  verify  by 
multiplying  2678  by  3426,  and  adding  1470  to  the  product,  as 
the  table  of  operations  shows.  (The  four  operations  which  we 
have  just  performed  in  this  division,  conduct  to  the  same  result 
as  if  we  had  subtracted  successively  from  the  dividend,  3000 
times,  then  400  times,  then  20  times,  then  6  times  the  proposed 
divisor.) 

Second  Example. 

Given,  to  divide  42206581591,  by  569874. 
42206581591 i 569874 
2315401  74063 


3590559 
1713151 


3529 
Placing  mentally  four  zeros,  then,  five  zeros,  to  the  right  of  the 
divisor,  we  obtain  two  products,  5698740000,  and  56987400000, 
which  contain  the  dividend  between  them;  which  proves  that 
the  quotient  sought  is  itself  comprised  between  10,000  and 
100,000,  or  is  composed  of  5  figures,  of  which  the  first  to  the 
left  expresses  the  tens  of  thousands.  The  two  first  figures  on  the 
left  of  this  quotient,  74,  which  we  have  placed  below  the  divisor, 


DIVISION.  49 

are  found  without  difficulty,  as  in  the  first  example.  But,  having 
arrived  at  the  remainder,  35,905,  if,  in  order  to  form  the  third 
partial  dividend,  which  is  to  furnish  the  hundreds  figure  of  the 
total  quotient,  we  bring  down  to  the  right  of  this  remainder  the 
next  figure,  5,  of  the  dividend,  we  obtain  359055,  a  number  less 
than  the  divisor,  which  proves  that  the  quotient  has  no  hundreds. 
We  must  then  place  a  0  to  the  right  of  the  two  first  quotients, 
then  bring  down  the  next  figure,  9,  to  the  right  of  359055 ;  we 
thus  have  a  fourth  partial  dividend,  3590559,  which  we  divide 
by  569874,  in  order  to  obtain  the  tens  of  the  quotient.  Con- 
tinuing the  operation,  we  find,  finally,  the  total  quotient,  74063, 
with  the  remainder,  3529. 

GrENERAL  RuLE. 

36.  In  order  to  divide  any  two  entire  numbers  whatever ,  one 
hy  another^  write  the  divisor  to  the  right  of  the  dividend ;  sepa- 
rate them  hy  a  vertical,  then  draw  a  horizontal  line  helow  the 
divisor.  This  done,  take  on  the  left  of  the  dividend  the  number 
of  figures  necessary  and  sufficient  to  contain  the  divisor ;  you 
obtain  thus  a  first  partial  dividend,  composed  either  of  as  many 
figures,  or  one  more,  than  there  are  in  the  divisor.  See  how 
many  times  this  partial  dividend  contains  the  divisor,  and  larite 
the  resulting  quotient  under  the  divisor  ;  multiply  the  divisor  by 
this  figure,  and  subtract  the  product  from  the  first  partial  divi- 
dend. 

Bring  down  to  the  right  of  the  remainder,  the  next  figure  of 
the  dividend,  which  gives  a  second  partial  dividend.  See  in  the 
same  manner  how  many  times  this  second  partial  dividend  con- 
tains the  divisor,  and  write  this  new  quotient  to  the  right  of  the 
first.  Multiply  the  divisor  by  this  second  quotient,  and  subtract 
the  product  from  the  second  partial  dividend. 

Bring  doivn  to  the  right  of  this  second  remaiiider,  the  next 
figure  of  the  dividend,  which  gives  a  third  partial  dividend^ 
upon  which  you  operate  as  upon  the  preceding. 

Continue  this  series  of  operations,  until  you  have  brought  down 
the  last  figure  of  the  dividend,  taking  care  at  each  operation  to 
5 


50  DIVISION. 

write  the  quotient  which  you  obtain  to  the  right  of  the  preceding, 
(in  order  to  give  to  the  former  their  jyroper  value^. 

If  it  happens  after  having  brought  doivn  a  figure,  that  you 
obtain  a  partial  dividend  less  than  the  divisor,  place  a  0  in  the 
quotient,  and  then  bring  down  a  new  figure  to  form  a  new  'partial 
dividend.  When,  after  all  these  operations,  we  arrive  at  a  re- 
mainder, 0,  the  dividend  is  said  to  be  exactly  divisible  by  the 
divisor  ;  if  the  remainder  is  not  0,  we  add  it  in  the  verification 
to  the  product  of  the  divisor  by  the  quotient. 

37.  From  the  nature  of  the  method,  we  deduce  the  following 
consequences : 

1st.  No  partial  division  can  give  a  quotient  greater  than  9,  nor 
ought  to  lead  to  a  remainder  less  than  the  divisor. 

2d.  The  first  figure  on  the  left  of  the  quotient  expresses  units 
of  the  same  order  as  those  expressed  by  the  first  figure  on  the 
right  of  the  first  partial  dividend ;  and  consequently  the  quotient 
contains  one  more  figure  than  the  rest  of  the  dividend,  after  the 
first  partial  dividend  has  been  separated.  In  other  terms,  the 
number  of  figures  of  the  quotient  is  either  the  difference  between 
the  number  of  figures  in  the  dividend  and  the  number  of  figures 
in  the  divisor,  or  this  difference  augmented  by  unity. 

Particular  Cases  of  Division. 

EemarJc.  —  When  one  of  the  terms  of  a  division  to  be  per- 
formed, or  both  of  them,  are  terminated  by  zeros,  we  can  simplify 
the  general  method. 

We  will  examine,  specially,  the  case  in  which  the  divisor  alonS 
is  terminated  by  zeros,  as  the  same  rule  of  simplification  can  be 
applied  to  all  other  cases,  a  single  one  excepted. 

1st.  Given,  47543296  to  be  divided  by  690000. 

General  method.  Particular  method. 

47543296  I  690000  4754  1 3296 1  69 

6143296 1 68  614 1  1 68 


623296  Rem.     62        True  rem.  623296 


DIVISION.  51 

The  rule  to  be  followed  is, — suppress  the  zeros  which  terminate 
the  divisor,  and  separate  on  the  right  of  the  dividend  as  many 
figures  as  there  are  zeros  on  the  riojht  of  the  divisor.  There  re- 
mains  then  to  be  divided,  4754  by  69 ;  perform  this  division 
after  the  ordinary  method.  The  quotient  obtained,  68,  is  the 
true  quotient  of  the  division  of  the  given  numbers.  Write  after 
the  corresponding  remainder,  62,  the  figures  3296  of  the  divi- 
dend which  have  been  separated  to  the  right,  and  you  have 
623296  for  the  total  reinainder  of  the  division. 

This  manner  of  operating  rests  on  the  following  : — We  observe, 
first,  that  the  69  tens  of  thousands  of  the  primitive  divisor  are 
contained  in  the  4754  tens  of  thousands  of  the  dividend,  the 
same  number  of  times  that  69  simple  units  are  contained  in  4754 
simple  units.  Thus,  the  quotient  of  4754  by  69  ought  to  be 
identical  with  the  quotient  of  the  division  of  the  two  given 
numbers. 

In  the  second  place,  the  remainder  of  the  division  of  4754  by 
69,  being  less  than  the  divisor,  69,  it  follows  that  this  remainder, 
followed  by  the  figures  to  the  right  of  the  dividend,  3296,  is  less 
also  than  the  divisor  followed  by  four  zeros,  or  690000 ;  then, 
623296  expresses  the  true  remainder  of  the  division  of  the  two 
numbers,  47543296  and  690000,  (as  620000  would  be  the  re- 
mainder, if  the  dividend  were  47540000.  So  623296  must  be 
the  true  remainder  in  the  case  above). 

2d.  The  case  in  which  the  dividend  alone  is  terminated  by 
zeros,  does  not  give  place  for  any  simplification.  Nevertheless, 
if  the  part  to  the  left  of  these  zeros  is  to  contain  the  divisor 
exactly,  we  could  first  cut  off  the  zeros ;  then,  after  having  ob- 
tained the  quotient  of  the  division  of  the  part  on  the  left  by  the 
divisor,  we  would  write  to  the  right  of  this  quotient  the  zeros 
which  terminate  the  dividend. 

EXAMPLE. 

Given,  375000  to  be  divided  by  125.  The  division  of  375  by 
125,  giving  3  for  exact  quotient,  the  quotient  demanded  is  3, 


52  DIVISION. 

followed  by  three  zeros  of  the  dividend,  or  3000.    But  this  sim- 
plification is  of  no  importance. 

3d.  The  same  number  of  zeros  may  be  on  the  right  of  both 
dividend  and  divisor.     In  this  case,  we  suppress  the  zeros  in  the 
two  terms  of  the  division ',  then,  after  having  divided  the  two 
parts  on  the  left,  one  by  the  other,  we  write  after  the  remainder 
thus  obtained,  the  zeros  on  the  right  of  the  dividend. 
Given,  to  divide  5679800  by  8600. 
56798|00  I  86100 
519"         1660" 
Rem.        38  True  rem,  3800 

This  mode  of  operating  is  a  repetition  of  the  first  case,  with 
the  sole  diff*erence  that  the  remainder  of  the  division  of  56798 
by  86,  or  38,  is  to  be  followed  by  the  zeros  which  terminate  the 
primitive  dividend,  instead  of  being  followed  by  significant 
figures. 

4th.  Fewer  zeros  on  the  right  of  the  dividend  than  on  the 
right  of  the  divisor. 

Given,  to  divide  68235947000  by  547600000. 
68235914700015476 
13475  1124 

25239 


Eem.  3335  True  rem.  333547000 

This  case  is  compounded  of  the  first  and  third. 
5th.  More  zeros  on  the  right  of  the  dividend  than  on  the  right 
of  the  divisor. 

Given,  to  divide  25036900000  by  875000. 
25036900[000  I  875 
7536  1 28613 

~5369 
"1190 
3150 
Eem.     525  '  True  rem.  525000 

This  case  is,  properly  speaking,  nothing  more  than  a  particular 
case  of  the  first  case. 


DIVISION.  53 

General  Remarh.  —  As  in  the  three  first  operations  of  arith- 
metic, the  calculations  are  performed  by  commencing  on  the  right, 
it  is  natural  to  ask  why,  in  division,  we  commence  on  the  con- 
trary on  the  left. 

In  order  to  answer  this  question,  we  must  observe  that  the 
dividend,  being  the  sum  of  the  partial  products  of  the  divisor, 
by  the  units,  tens,  hundreds,  &c.,  of  the  quotient,  all  these  par- 
tial products  are  mingled  one  with  another ;  so  that  it  is  impos- 
sible to  commence  by  separating  out  the  products  by  the  units, 
by  the  tens,  &c. ;  while,  by  the  established  method  we  determine 
at  once  in  what  part  of  the  dividend  the  product  by  the  highest 
units  is  found,  and  then  we  obtain  the  figure  of  these  highest 
units ;  then  we  arrive  at  the  figure  of  the  units  of  the  order  im- 
mediately below  the  first,  and  thus  with  the  rest. 

Two  Examples  for  Exercise. 

1st.  12187610837  to  be  divided  by  15619. 

Quotient,  780306;  remainder,  11423. 
2d.   2487623393304  to  be  divided  by  5076078. 

Quotient,  490068 ;  remainder,  0. 

Verification  of  Multiplication  and  of  Division. 
It  has  been  established  in  (No.  27),  that  we  are  naturally  led 
by  the  definition,  even  of  division,  to  make  the  proof  of  Mul- 
tiplication by  Division,  and  that  of  Division  by  Multiplication  ; 
and  we  have  given  in  the  course  of  the  exposition  of  the  method, 
the  means  of  performing  this  operation.  But  we  will  show  later 
more  expeditious  methods  of  making  these  verifications. 

88.  We  will  now  give  some  uses  of  multiplication  and  divi- 
sion. 

Question  1st.  —  Required  the  price  of  2564  yards  of  a  piece 
of  masonry,  of  which  each  yard  costs  47  dollars. 

Since  each  yard  costs  47  dollars,  it  is  clear  that  by  repeating 
this  value  2564  times,  we  will  have  the  price  of  the  2564  yards. 
Thus,  it  suffices  to  form  the  product  of  47  by  2564,  or  rather 


54  DIVISION. 

the  product  of  2564  by  47 ;  and  this  product  will  express  the 
number  of  dollars  required. 

We  give  the  operation  and  its  proof  by  division. 
2564 
47 


17948 
10256  188 


120508 

47 

265 
300 

2564 

120508  000 

The  2564  yards  cost  $120,508. 
Question  2d. — One  yard  of  a  piece  of  masonry  cost  $39. 
Required  how  many  yards  can  be  built  for  $8395. 

It  is  evident  that  as  many  times  as  39  will  be  contained  in 
8395,  so  many  yards  can  be  constructed  for  the  price.  Thus,  it 
suffices  to  divide  8395  by  39 ;  and  the  quotient  will  be  the  re- 
quired number  of  yards. 

8395  I  39 Proof,     215 

59      12151^  _39 

205  1935 

10  645 

10 
8395 
As  we  obtain  215  for  quotient,  and  10  for  remainder,  it  is  ne- 
cessary to  know  the  use  which  we  are  to  make  of  this  remainder. 
Let  us  observe,  that  if  the  dividend  contained  $10  less,  it 
would  be  the  product  of  39  by  215.    Thus,  the  number  of  yards 
demanded  would  be  215 ;  but,  as  we  have  10  dollars  more,  we 
have  to  determine  the  part  or  fraction  of  a  yard  which  can  be 
constructed  for  these  10  dollars. 

Now,  with  one  dollar,  we  would  evidently  have  g'^th  of  a  yard, 
since  we  could  build  the  whole  yard  for  $39;  then,  with  $10, 
we  must  have  10  times  -g^^,  or  (No.  8),  \^.  Thus,  215  yards, 
and  -J£ths  of  a  yard,  is  the  result  required. 

Such  is,  in  general,  the  use  which  we  have  to  make  of  the  re- 
mainder of  a  division,  when  in  performing  the  operation  we  are 
resolving  a  question  relating  to  concrete  numbers. 


DIVISION.  .     55 

We  conceive  the  unit  of  the  quotient  (the  nature  of  whicli  is 
always  determined  by  the  enunciation  of  the  question),  to  he 
divided  into  as  many  equal  parts  as  there  are  units  in  the  divi- 
sor', we  take  one  of  these  parts  as  tnany  times  as  there  are  units 
in  the  remainder ;  we  then  add  the  resulting  fraction  to  the  en- 
tire quotient  already  obtained. 

Question  3d. — Suppose  that  498  persons  have  to  divide  equally 
a  sum  of  $1,348,708.  •  Required  the  part  of  each  one. 
13487081498  2708 


3527 

1 2708i§| 

498 

4108 

21664 

124 

24372 
10832 
124 

1348708 

The  quotient  of  this  division  being  2708,  and  the  retiiainder 
124,  we  conclude  that  if  the  sum  to  be  divided  was  diminished 
by  124  dollars,  each  person  would  have  for  his  portion  2708 
dollars.  But,  as  the  given  sum  contains  $124  more  than  the 
product  of  2708  by  498,  it  follows  that  each  person  ought  to 
have  $2708,  and  a  part  of  $124.  In  order  to  form  an  idea  of 
this  part,  we  can  at  first  consider  the  number  124  as  a  whole, 
which  it  is  necessary  to  divide  into  498  equal  parts ;  and  one  of 
these  parts  is  the  fraction  which  is  to  complete  the  quotient :  but 
it  is  simpler  to  conceive  the  unit  (No.  8),  1  dollar,  to  be  divided 
into  498  equal  parts,  and  to  take  124  of  these  parts,  which  gives 


1  2  4 

4"5H 


for  the  fraction  to  be  added  to  the  entire  quotient. 


39.  N.  B.  —  This  last  example  leads  us  to  a  remark  of  which 
we  will  often  make  use ;  it  is,  that  to  divide  a  number,  124,  into 
498  equal  parts,  is  to  take  124  times  the  498th  part  of  unity. 
For,  if  instead  of  124,  we  had  simply  to  divide  1  into  498  equal 
parts,  each  part  would  be  (No.  8),  ^^^ )  but,  as  the  number  to 
be  divided  is  124  times  greater,  the  result  of  the  division  ought 
to  be  124  times  greater,  or  equal  to  124  times  ^^j,  or  equal  to 


50     .  DIVISION. 

|||.  In  general,  to  divide  a  numher  into  as  many  equal  parts 
as  there  are  units  in  another,  is  the  same  thing  as  to  divide  unity 
into  as  many  equal  parts  as  there  are  units  in  the  second  number, 
and  to  take  one  of  these  parts  as  many  times  as  there  are  units 
in  the  first. 

40.  From  the  two  propositions  demonstrated  in  Nos.  25  and 
26,  we  deduce  some  consequences  which  it  is  well  to  make  known, 
as  they  are  of  continual  use  in  arithmetic. 

We  observe,  first  of  all,  that  according  to  the  definitions,  even 
of  the  multiplication  and  division  of  entire  numbers,  we  render 
an  entire  number  as  many  times  greater  or  smaller  as  there  are 
units  in  another,  in  multiplying  or  dividing  the  first  number  by 
the  second. 

Thus,  when  we  multiply  24  by  6,  the  product  which  we  obtain 
is  6  times  as  great  as  24,  since  it  results  from  the  addition  of  6 
numbers  equal  to  24.  In  the  same  manner,  if  we  divide  24  by 
6,  the  quotient  is  6  times  as  small  as  24,  since  this  quotient,  re- 
peated 6  times,  reproduces  24. 

This  established,  we  say,  first,  that  if  in  a  multiplication  we 
render  the  multiplicand  or  multiplier  a  certain  number  of  times 
greater  or  smaller,  the  product  is,  by  this  change,  rendered  the 
same  number  of  times  greater  or  smaller. 

Given,  for  example,  to  multiply  47  by  6,  and  suppose  that,  in- 
stead of  performing  this  operation,  we  multiply  47  by  24,  which 
is  4  times  as  great  as  6 ;  since,  according  to  what  has  been  said 
in  (25),  to  multiply  47  by  24,  is  the  same  as  to  multiply  47  by  6, 
and  the  product  by  4,  it  follows  at  once  that  the  product  of  47 
by  24,  equals  4  times  the  product  of  47  by  6;  (t*.  e.)  is  4 
times  as  great. 

Reciprocally,  the  product  of  47  by  6  (the  fourth  of  24),  being 
4  times  smaller  than  the  product  of  47  by  24,  it  follows,  that  if 
we  render  the  multiplier  4  times  as  small,  or  if  we  divide  it  by  4, 
the  product  is  rendered  4  times  as  small  by  this  change. 

We  have  seen,  moreover  (26),  that  in  a  multiplication  of  two 
factors.,  we  can  invert  the  order  of  the  factors;  then,  what  wo 


DIVISION.  5 1 

have  just  said  with  reference  to  the  multiplier,  applies  equally  to 
the  multiplicand ;  then, 

It  results  from  this,  that  we  do  not  change  a  product,  in  ren- 
dering the  multiplicand  a  certain  number  of  times  greater,  pro- 
vided we  render  at  the  same  time  the  multiplier  a  certain  number 
of  times  less,  (^.  e.)  by  multiplying  one  factor  by  a  certain  num- 
ber, and  dividing  the  other  factor  by  the  same  number. 

It  is  upon  this  last  consequence  that  we  found  a  method  which 
is  sometimes  employed  to  verify  multiplication. 

Given,  to  multiply  347  by  72.  To  multiply  347  by  72,  is  to 
multiply  twice  347,  or  694,  by  the  half  of  72,  or  by  36.  Thus, 
after  having  multiplied  347  by  72,  we  can  then  multiply  694  by 
36;  and,  if  the  operations  are  correct,  we  ought  to  find  the 
same  result. 

Now,  since  in  division  the  dividend  is  a  product,  of  which  the 
divisor  and  the  quotient  are  two  factors,  it  follows  that  if  we 
multiply  or  divide  the  dividend  by  a  certain  entire  number,  the 
quotient  is  by  this  change  multiplied  or  divided  by  the  same  en- 
tire number. 

For,  as,  after  this  change,  the  quotient,  multiplied  by  the  divi- 
sor, must  produce  a  dividend  a  certain  number  of  times  greater 
or  less  than  the  first  dividend,  it  follows  necessarily,,  the  divisor 
remaining  the  same,  that  the  quotient  must  be  the  same  number 
of  times  greater  or  less. 

On  the  contrary,  if,  without  altering  the  dividend,  we  render 
the  divisor  a  certain  number  of  times  greater  or  smaller,  the 
quotient  is  thereby  rendered  the  same  number  of  times  smaller 
or  greater.  This  is  the  sole  admissible  hypothesis,  in  order  that 
the  multiplication  may  give  the  same  product  or  the  same  divi- 
dend. 

Then,  by  multiplying  o^j^  dividing  the  dividend  and  the  divisor 
by  the  same  number,  we  do  not  change  the  quotient ;  since,  if, 
by  the  change  of  the  dividend,  we  multiply  or  divide  the  quotient 
by  a  certain  number,  the  second  change  renders  it  the  same 
number  of  times  smaller  or  greater.  Thus,  the  compensation 
leaves  it  the  same. 


68  DIVISION. 


Exercises  on  tliis  Chapter. 

1.  Enunciate  the  number  10030047089500476. 

2.  Enunciate  the  same  number,  the  middle  figure  being  left 
out. 

3.  How  many  figures  in  a  number,  the  first  figure  on  the  left 
of  which  expresses  hundreds  of  septillions  ? 

4.  What  is  wanting  in  the  number  2047035007,  in  order  to 
form  unity,  followed  by  as  many  zeros  as  there  are  figures  in  the 
number  ? 

5.  Being  given  to  subtract  58900564  from  62080347,  if  we 
substitute  for  the  smaller  number  that  which  would  make  it 
unity  followed  by  8  zeros,  and  if  we  add  this  complement  to  the 
greater  number,  what  must  we  then  do  to  obtain  a  result  equal 
to  that  which  we  obtain  by  the  direct  operation  ? 

6.  The  day  being  composed  of  24  hours,  the  hour  of  60  mi- 
nutes, the  minute  of  60  seconds ;  required  how  many  seconds 
there  are  in  the  year,  which  we  suppose  to  have  exactly  365 
days. 

7.  Required  what  changes  would  be  made  in  the  product  of 
67084  by  3769,  by  supposing,  1st,  That  the  multiplier  is  aug- 
mented by  10,  the  multiplicand  remaining  the  same  ?  2d,  That 
the  multiplicand  is  augmented  by  10,  the  multiplier  remaining 
the  same?  3d,  That  the  two  factors  are  simultaneously  aug- 
mented by  10  units  ? 

8.  What  is  the  population  of  a  county  containing  16537 
square  miles,  each  square  mile  averaging  45  inhabitants  ? 

9.  The  light  of  the  sun  reaches  the  earth  in  8  minutes  and  13 
seconds,  and  the  distance  is  95000000  miles.  How  far  does  the 
light  travel  in  one  second  ? 


FRACTIONS.  59 


CHAPTER  II. 

FRACTIONS. 

41.  We  have  already  seen  (Nos.  1  and  8),  what  a  fraction  is, 
and  what  idea  we  are  to  form  of  it.  We  distinguish  always  two 
terms  in  a  fraction,  the  denominator  and  the  numerator.  The 
denominator  indicates  into  how  many  equal  parts  unity  is  divided, 
and  the  numerator  how  many  of  these  parts  are  taken.  These 
two,  taken  together,  constitute  the  fraction. 

Thus,  in  the  fraction  |,  which  we  call  three-fourths,  4  is  the 
denominator,  and  shows  that  the  unit  is  divided  into  4  equal 
parts ',  3  is  the  numerator,  and  indicates  that  we  take  3  of  these 
parts.  In  the  same  manner,  the  fraction  ii,  eleven-twelfths, 
expresses  that  the  unit  is  divided  into  12  equal  parts,  and  that 
we  take  eleven  of  them.  We  have  seen,  also,  that  such  a  fraction 
as  i|  is  equivalent  to  the  15th  part  of  the  whole  number  ex- 
pressed by  13  ',  that  is  to  say,  a  fraction  can  also  be  considered 
as  the  quotient  of  its  numerator  divided  by  its  denominator,  so 
that  thirteen  times  the  15th  part  of  unity,  or  thirteen-fifteenths, 
and  the  j&fteenth  part  of  thirteen,  or  thirteen  divided  by  fifteen, 
are  identical  expressions. 

This  last  point  of  view  leads  us  naturally  to  consider  fractional 
expressions,  such  as  \^,  f |,  |J,  ....  of  which  the  numerator  is 
greater  than  the  denominator. 

Now,  these  expressions  are  easily  comprehended,  as  they  result 
from  the  division  of  the  numbers  19,  23,  47,  respectively  into  6, 
12,  15,  equal  parts. 

But  how  can  y  express  19  times  the  6th  part  of  unity  ? 

For  this  we  conceive  that  we  have  four  principal  units,  of 
which  each  one  is  divided  into  6  sixths ;  then,  in  order  to  form 
19,  or  6x3  +  1  of  them,  we  take  the  18  sixths,  of  which  the 


60  FRACTIONS. 

three  first  principal  units  are  composed,  and  add  to  them  one  of 
the  parts  of  the  fourth  principal  unit. 

We  obtain  thus  19  sixths,  or  ^^. 

By  extending  this  principle,  we  can  place  unity  or  any  entire 
number  under  a  fractional  form. 

Thus,  1  can  be  written  j|,  |f,  &c. 

In  the  same -manner,  10,  14,  25,  &c.,  can  be  written  ^-f,  y, 
2/,  &c. 

42.  From  the  definition  which  we  have  just  given  of  the 
numerator  and  denominator  of  a  fraction,  the  following  conse- 
quences obviously  result. 

1st.  If  we  multipli/  or  divide  the  numerator  of  a  fraction  hy  a 
numherj  the  denominator  remaining  the  sam.e,  the  new  fraction 
will  he  this  number  of  times  greater  or  less  than  the  first. 

For,  when  we  multiply  the  numerator  by  2,  3,  4,  ...  .  we  in- 
dicate thereby,  that  we  take  2,  3,  4,  ...  .  more  parts;  and  as 
the  parts  are  the  same,  the  new  fraction  is  2,  3,  4,  ...  .  times 
greater.  Thus,  let  the  fraction  be  -/^ ;  it  is  clear  that  ^|,  ^-|, 
||-,  ....  are  fractions  2,  3,  4,  ...  .  times  greater  than  the  first. 
Again,  in  dividing  the  numerator  by  2, 3, 4  ....  we  take  2, 3, 4  ... . 

times  fewer  parts  than,  &c Thus,  ^^g,  -^-g,  are  2,  3,  times 

smaller  than  -^^. 

2d.  If  we  multijpli/  or  divide  the  denominator  of  a  fraction 
hy  a  numher,  the  numerator  remaining  unchanged,  we  divide  or 
multiply  the  fraction  hy  this  numher 

For,  when  we  multiply  the  denominator  by  2, 3, 4  .  ...  we  indi- 
cate that  the  unit  is  divided  into  parts  2,  3,  4  ...  .  times  more 

parts.  Each  of  these  parts  is  then  2,  3,  4 times  smaller;  and  as 

we  take  always  the  same  number  of  these  parts,  it  follows  that 
the  fraction  is  2,  3,  4  ...  .  times  smaller. 

If  we.  divide  the  denominator  by  2,  3,  4  ...  .  the  unit  is 
divided  into  2,  3,  4  ...  .  times  fewer  parts ;  each  one  of  theso 
parts  is  then  2,  3,  4  ...  .  times  greater;  therefore,  &c. 

3d.  We  do  not  change  the  value  of  a  fraction  hy  multiplying 
or  dividing  its  two  terms  hy  the  same  numher. 


FRACTIONS.  61 

For  the  change  made  in  the  numerator  renders  the  fraction  a 
certain  number  of  times  greater  or  smaller ;  but  the  same  change 
made  in  the  denominator  renders  it  on  the  contrary  the  same 
number  of  times  smaller  or  greater;  then  one  change  compen- 
sates the  other,  and  the  value  of  the  fraction  is  not  change'd. 

Thus,  the  fractions  |,  -f^,  ||,  H,  ....  are  all  equivalent  to  the 
fraction  |,  since  they  result  from  the  multiplication  of  each  of 
the  tei-ms  of  the  latter  by  2,  3, 4,  5.  In  the  same  way,  the  fraction 
-||  is  equal  to  the  fraction  j|,  or  j%,  or  |,  since  we  obtain  these 
last  by  dividing  the  two  terms  of  ||  by  2,  3,  4. 

These  different  propositions  can  also  be  considered  as  conse- 
quences of  the  second  manner  of  viewing  a  fraction  (see  No.  41), 
and  the  principles  established  (No.  40),  in  division. 

43.  As  the  third  proposition  is  of  continual  application,  we 
will  give  a  demonstration  of  it,  direct  and  independent  of  the 
two  first. 

Take,  for  example,  the  fraction  |,  and  multiply  the  two  terms 
5  and  8  by  3,  which  gives  ||.  "We  have  to  prove  that  this  last 
fraction  is  equivalent  to  the  first. 

For,  the  principal  unit  being  divided  at  first  into  eight  equal 
parts,  let  us  divide  each  eighth  into  three  equal  parts ;  the  unit 
is  thus  divided  into  twenty-four  equal  parts. 

Each  eighth  equals,  then,  three  twenty-fourths,  and  five-eighths 
equal  five  times  three,  or  fifteen  twenty-fourths ;  that  is  to  say, 
the  fractions  |  and  ^|  have  absolutely  the  same  value. 

We  would  demonstrate,  in  the  same  manner,  that  the  fractions 
i^  and  |g,  the  latter  of  which  is  formed  by  multiplying  the  two 
terms,  11  and  12,  of  the  first,  by  5,  are  equal. 

As  reciprocally  we  pass  from  the  fraction  i|  to  the  fraction  |, 
by  taking  the  third  of  each  term,  and  from  the  fraction  |J,  to 
the  fraction  |^,  by  taking  the  fifth  of  the  two  terms  of  the  for- 
mer, we  can  conclude  that  a  fraction  does  not  change  its  value 
when  we  multiply  or  divide  its  two  terms  hy  the  same  number. 

Let  us  pass  to  the  different  operations  which  we  may  have  to 
perform  on  fractions  in  the  resolution  of  a  question,  the  data  of 
which  are  fractions  or  fractional  numbers. 
G 


62  FRACTIONS. 

But,  before  explaining  tlie  four  fundamental  operations,  we 
will  make  known  two  transformations  of  frequept  use  in  the 
calculus  of  fractions. 

Reduction  op  Fractions  to  the  same  Denominator. 

44.  This  transformation  has  for  its  object  to  reduce  two  or  more 
fractions  J  having  different  denominator's ,  to  the  same  denomina- 
tor. Now,  the  principle  that  we  do  not  change  the  value  of  a 
fraction  by  multiplying  its  two  terms  by  the  same  number,  fur- 
nishes a  simple  means  of  effecting  this  transformation. 

Let,  for  example,  |  and  |  be  the  fractions  which  are  to  be  re- 
duced to  the  same  denominator. 

If  we  multiply  the  two  terms,  3  and  4,  by  7,  the  denominator 
of  the  second,  and  the  two  terms,  5  and  7,  of  the  second,  by  4, 
the  denominator  of  the  first,  there  will  result  |^  and  |g  for  the 
two  fractions  required. 

These  fractions  have  the  same  value  as  the  fractions  proposed 
according  to  the  principle  of  (43).  Again,  they  have  necessarily 
equal  denominators,  since  each  one  of  them  comes  from  the  mul- 
tiplication of  the  two  primitive  denominators,  4  and  7,  by  each 
other. 

Again,  given  the  fractions,  ^j  %,  tt,  to  be  reduced  to  the  same 
denominator. 

Multiply  the  two  terms,  4  and  7,  of  the  first  fraction,  by  88, 
product  of  the  denominators,  8  and  11,  of  the  second  and  third; 
then  the  two  terms,  5  and  8,  of  the  second,  by  77,  product  of 
the  denominators,  7  and  11,  of  the  first  and  third;  finally,  the 
two  terms,  6  and  11,  of  the  third,  by  56,  product  of  the  deno- 
minators of  the  first  and  second.  We  will  thus  obtain  the  new 
fractions,  |f|,  |f |,  |f |. 

These  fractions  have  the  same  value  as  the  primitive  fractions, 
and  their  denominators  are  necessarily  the  same,  since  each  one 
of  them  is  the  product  of  the  denominator  of  each  fraction  by 
the  product  of  the  two  other  denominators. 


FRACTIONS.  63 

General  Rule.  —  In  order  to  reduce  any  numher  whatever 
of  fractions  to  the  same  denominator ,  multiply  successively  the 
two  terms  of  each  one  of  them  hy  tM  product  of  the  denominators 
of  the  other  fractions. 

We  will  show  the  method  of  applying  this  rule  in  practice. 

Let  the  fractions  be  |,  y\,  f§,  ||,  and  ||. 

For  greater  simplicity  we  arrange  the  operation  thus : 

3  7  JO  23  29 

8  11  13  3?  5  4  3 


153725    111800    94600    49192    28600 

461175  782600  9  4  6  0  0  0_  1131416  829400 

r22^B0U?        122^500?       123^500^       T32]5B00>       TS-JUgOU* 


After  having  formed  the  product  of  the  five  denominators,  8, 
11,  13,  25,  and  43,  which  gives  for  the  common  denominator  of 
the  transformed  fractions,  1229800,  we  divide  successively  this 
product  by  each  one  of  the  denominators,  and  we  obtain  the  five 
quotients,  153725,  111800,  94600,  49192,  28600,  which  we 
place  respectively  below  the  five  proposed  fractions ;  after  which, 
we  multiply  the  numerator  of  each  fraction  by  the  quotient  which 
corresponds  to  it ;  and  we  obtain  thus  the  difi"erent  numerators. 

As  to  the  common  denominator,  it  is,  as  we  have  said  above, 
equal  to  1229800. 

The  reason  of  this  manner  of  proceeding  is  easily  perceived, 
for  the  number,  1229800,  being  the  product  of  the  five  denomi- 
nators, the  quotient,  153725,  of  the  division  of  1229800  by  8, 
expresses  necessarily  the  product  of  the  four  other  denominators, 
11,  13,  25,  43. 

In  the  same  manner,  111800,  being  the  quotient  of  the  divi- 
sion of  1229800,  by  the  second  denominator,  11,  is  equal  to  the 
product  of  the  four  other  denominators,  8,  13,  25,  and  43 ;  and 
the  same  reasoning  applies  to  the  other  quotients.  This  method 
is,  moreover,  much  more  expeditious,  than  if,  for  each  fraction, 
we  performed  the  multiplication  of  the  denominators  of  the  four 
others.  But  it  is  only  really  advantageous  when  there  are  more 
than  three  fractions  to  be  reduced  to  the  same  denominator. 


64  FRACTIONS. 

45.  There  is  a  case  in  which  the  reduction  to  the  same  deno- 
minator can  be  performed  in  a  very  simple  manner^;  that  is,  when 
the  greatest  of  the  denominators  is  exactly  divisible  by  each  one 
of  the  others. 

Let  the  fractions  be,  for  example, 

2  3  5  7  23 

3  4  g  1'2  3  5 

12       9         6  3  1 

24  27  30  21  23 

3BJ       "3  6?  3  6?  3gJ  3B' 

It  is  easy  to  see  that  36,  divisible  by  itself,  is  also  divisible  by 
each  one  of  the  four  other  denominators,  3,  4,  6,  and  12. 

This  being  fixed,  we  effect  successively  these  divisions,  and 
place  the  quotients,  12,  9,  6,  31,  below  the  four  first  fractions; 
after  which,  we  multiply  the  numerator  of  each  one  of  them  by 
the  quotient  which  corresponds  to  it;  the  fraction,  ||,  remains 
as  it  was,  and  all  the  fractions  are  reduced  to  the  denominator^  36. 

Sometimes,  although  the  greatest  denominator  is  not  divisible 

by  all  the  others,  we  perceive  that,  by  multiplying  it  by  2,  3, 4 

we  obtain  a  product  exactly  divisible  by  all  the  denominators. 
This  afibrds  us,  likewise,  a  means  of  simplification. 

Let  the  fractions  be, 


2 
4 

i 

H 

1  3 
IB 

1  7 
24 

II 

18 

9 

G 

4 

3 

2 

H 

fl 

^1 

52 

72- 

H 

H 

The  denominator,  36,  is  divisible  separately  by  4,  12,  and  18, 
but  is  not  divisible  by  8  nor  by  24 ;  but,  if  we  double  it,  we  ob- 
tain 72,  a  number  exactly  divisible  by  each  one  of  the  denomi- 
nators. 

This  being  fixed,  we  form  the  quotients  of  72  by  each  one  of 
the  denominators,  and  place  them  respectively  below  the  frac- 
tions ;  we  then  multiply  the  numerator  of  each  one  of  them  by 
the  quotient  which  corresponds  to  it;  all  these  fractions  will 
have  72  for  common  denominator. 


FRACTIONS.  65 


Formation  of  the  Least  Common  Denominator  of 
Several  Fractions. 

46.  The  simplifications  which  we  have  just  explained,  require 
some  practice  to  see  when  they  can  be  applied ;  but  there  is  a 
direct  means  of  obtaining,  in  all  cases,  the  Least  Common  De- 
nominator of  several  fractions. 

To  do  this,  we  must  find  the  least  common  multiple  of  the  de- 
nominators; that  is,  the  least  number  divisible  by  all  of  them. 
To  do  this,  decompose  the  numbers  into  their  smallest  possible 
factors ;  that  is,  prime  factors,  or  factors  divisible  only  by  them- 
selves and  unity.  Then  form  the  product  of  all  these  prime 
factors,  common  or  not  common,  to  the  numbers.  We  obtain 
thus  a  result,  evidently  divisible  by  all  the  numbers ;  and  it  is, 
besides,  the  smallest  number  so  divisible ;  for,  any  number  con- 
taining one  of  the  prime  factors  a  smaller  number  of  times 
than  one  of  the  given  numbers,  would  not  be  divisible  by  that 
one  of  these  numbers  which  contained  thfs  factor  a  greater  num- 
ber of  times.  (A  more  thorough  discussion  of  this  we  will  give 
under  the  chapter  on  the  properties  of  numbers). 

Applying  the  above  to  the  last  example,  we  have, 

4  8  12  18  24  36 

2.2         2.2.2       2.2.3        2.3.3       2.2.2.3      2.2.3.3 

Having  thus  arranged  the  numbers  and  their  prime  factors, 
we  see  that  2.2.2.3.3  is  evidently  the  least  common  multiple. 
Performing  the  multiplication,  we  obtain  72,  as  before. 

Let  the  fractions  be  for  a  new  example, 


11 

if 

1  7 

2  5 

37 

83 

29 

233 

450 

1? 

24 

2F 

44 

T45 

IIH 

the  numerators  of  which  do  not  contain,  at  least  apparently,  prime 
factors  (as  2.3.5  ....),  which  may  be,  at  the  same  time,  con- 
tained in  the  corresponding  denominators;  otherwise,  it  would 
be  necessary  to  suppress  these  factors  in  the  two  terms. 
6* 


66  FRACTIONS. 

Kecomposing  the  denominators,  we  find  for  results, 

3.5 1 2.3.3 1 2.2.2.3  1 2.2.7 1 2.2.11 1 2.2.5.7 1 7.5.5 1 2^2.2.2.2.5.3; 

which  gives  for  the  least  common  multiple, 

2.2.2.2.2.3.3.5.5.7.11,  or  554400; 

which  is  the  least  common  denominator  to  be  given  to  all  the 
fractions;  a  number  far  less  than  that  which  we  would  obtain  by 
applying  the  general  rule  in  No.  (44). 

Nothing  more  remains  now  but  to  determine  the  numbers  by 
which  we  are  to  multiply  the  numerators,  in  order  to  obtain  the 
numerators  of  the  new  fractions ;  and  for  this  it  is  necessary,  as 
we  have  already  seen,  to  divide  554400  by  each  one  of  the  given 
denominators. 

Kelations  of  Magnitude  among  Several  Fractions. 

We  have  here  some  applications  of  the  preceding  transforma- 
tions. 

47.  Question  1st.  —  Of  the  two  fractions^  |  and  ^^^  which  is 
the  greater  ? 

"VVe  cannot,  at  first  sight,  answer  this  question ;  because,  though 
on  the  one  hand  the  unit  in  the  second  fraction  is  divided  into  a 
greater  number  of  parts  than  in  the  first,  on  the  other  hand,  we 
take  more  of  these  parts,  since  the  numerator,  7,  is  greater 
than  3. 

But  we  remove  the  difiiculty  by  reducing  them  to  the  same 
denominator;  for  it  is  evident  that  of  two  fractions  which  have 
the  same  denominator,  the  greater  is  that  which  has  the  greater 
numerator.  This  reduction  effected,  we  obtain  |g  for  the  first 
fraction,  and  |J  for  the  second;  the  fraction,  |,  is  the  greater 
of  the  two. 

We  find,  in  the  same  manner,  that  of  the  three  fractions,  ^, 
A'  T^3'  ^^  greatest  is  f.^,  the  smallest  -{\)  for,  being  reduced  to 
the  same  denominator,  they  become,  respectively,  f^fj,  tVijVj 


FRACTIONS.  67 

We  could  equally  well  reduce  the  fractions  to  the  same  nume- 
rator (by  applying  to  the  numerators  what  has  been  said  concern- 
ing the  denominators)  ;  and  of  these  fractions  the  greatest  would 
be  that  which  would  have  the  smallest  denominator;  since,  the 
parts  being  greater,  we  take  the  same  number  of  them.  But  the 
first  method  has  the  advantage  of  making  known,  at  the  same 
time,  the  differences  which  exist  between  the  fractions,  compared 
two  and  two. 

48.  Question  2d.  —  What  change  do  we  produce  in  a  fraction^ 
hy  adding  the  same  number  to  its  two  terms  ? 

Let  the  fraction  be  -^^^  for  example,  to  both  terms  of  which 
we  add  6 ;  jf  is  the  resulting  fraction. 

If  now  we  reduce  these  two  fractions  to  the  same  denominator, 
the  first  becomes  ^||,  and  the  seeond  ^f  |.  The  proposed  frac- 
tion is  then  increased  in  value.  In  order  to  give  a  reason  for  this 
fact,  we  observe  that,  unity  being  equal  to  i|,  the  excess  of  unity 
above  ^^^  is  expressed  by  f^ ;  in  the  same  manner,  the  excess  of 
unity  above  ||  is  expressed  by  f^.  The  numerators  of  these 
two  differences  are  the  same,  which  should  be  the  case;  for,  18 
and  13,  having  been  formed  by  the  addition  of  6  to  the  two 
terms,  7  and  12,  it  follows,  that  there  is  the  same  difference  be- 
tween 18  and  13,  as  between  7  and  12.  But  the  difference,  -f^, 
is  necessarily  less  than  the  difference,  y^^,  since  the  first  denomi- 
nator is  the  greater,  and  the  numerators  are  equal;  then  the 
fraction,  ||,  differs  less  from  unity* than  the  fraction,  -^^^  conse- 
quently, the  first  is  greater  than  the  second. 

We  see,  moreover,  that  the  greater  the  number  added  to  the 
two  terms  of  the  fraction,  y^^,  the  smaller  the  difference  between 
unity  and  the  new  fraction ;  since  the  numerator  of  this  differ- 
ence, being  always  5,  the  denominator  becomes  greater  and 
greater.  As  this  same  reasoning  can  be  applied  to  every  other 
fraction,  we  can  draw  the  conclusion  that  if  to  the  two  terms  of 
a  fraction  we  add  thjB  same  number,  the  resulting  fraction  is 
greater  than  the  given  fraction ;  and  it  is  greater,  the  greater  the 
number  added. 


68  FRACTIONS. 

Conversely,  by  the  same  reasoning,  a  fraction  is  diminished  in 
value  when  we  subtract  the  same  number  from  its  two  terms. 

N.  B.  The  contrary  would  take  place,  if  the  fractional  number 
was  greater  than  unity,  as  i|. 

Adding  8  to  the  two  terms,  we  would  have  |  j  less  than  ||. 
For,  II  exceeds  unity  by  ^^  only,  while  i|  surpasses  unity  by 
■f^,  greater  than  ■^^. 

We  have  thought  it  necessary  to  enter  into  some  details  upon 
this  proposition,  in  order  to  prevent  beginners  from  confounding 
this  with  (43),  when  we  multiply  or  divide  the  two  terms  of  a 
fraction  by  the  same  number. 

Keduction  op  a  Fraction  to  its  Simplest  Terms. 

49.  It  happens  often,  in  the  calculus  of  fractions,  that  we  are 
led  to  a  fraction  expressed  by  large  numbers ;  now,  the  greater 
the  numerator  and  denominator,  the  greater  trouble  we  have  to 
form  a  just  idea  of  the  fraction. 

For  example,  the  fraction,  ||,  indicates,  that  we  must  divide 
unity  into  15  equal  parts,  and  take  12  of  these  parts.  But  12 
and  15  being,  at  the  same  time,  divisible  by  3,  if  we  perform  the 
divisions,  there  results  |,  a  fraction  equivalent  to  the  one  given ; 
then,  in  order  to  form  an  idea  of  it,  it  suffices  to  conceive  the 
unit  divided  into  5  equal  partg,  and  to  take  4  of  them,  which  is 
much  simpler.  When  then  we  have  a  fraction,  the  terms  of 
which  are  quite  large,  it  is  best  to  reduce  it,  if  possible,  to  a 
fraction  whose  terms  are  smaller. 

The  first  method  which  presents  itself  is  to  divide  the  two 
terms  by  the  numbers,  2,  3,  4  ....  as  long  as  that  is  possible. 

1st.  Let  the  fraction,  |J|,  he  given.  The  two  terms  of  this 
fraction  are  evidently  divisible  by  4 ;  and,  in  effecting  the  divi- 
sion, we  obtain  |^ ;  but  the  two  terms  of  this  are  divisible  by  9 ; 
and  this  new  division  gives  for  a  result,  |,  which  cannot  be  far- 
ther reduced. 


FRACTIONS.  69 

This  example  presents  no  difficulty ;  but  this  is  not  always  the 
case,  especially  when  the  two  terms  of  the  given  fraction  are 
composed  of  three  or  more  figures;  for  it  can  happen  that  a 
prime  factor  of  two  or  three  figures  is  common  to  the  two  terms 
of  the  fraction,  without  our  being  able  to  find  it  by  mere  inspec- 
tion. Hence,  we  see  the  necessity  of  having  a  general  method 
of  reducing  a  given  fraction  to  the  most  simple  expression  possi- 
ble. This  method  we  will  now  discuss.  It  is  called  the  method 
of  the  greatest  common  divisor. 

50.  Wo  commence  by  establishing  several  preliminary  no- 
tions. 

A  number  is  called  the  multiple  of  another  number,  when  it 
contains  it  a  certain  number  of  times,  as  we  have  already  seen. 

Reciprocally,  the  second  number  is  called  a  suhmultiple,  or  an 
aliquot  part,  or  simply  a  divisor  of  the  first. 

We  call  a  prime  number  a  number  which  is  only  divisible  by 
itself,  and  by  unity,  which  is  a  divisor  of  every  number.  Thus, 
2,  e3,  5,  7,  11,  13  ...  .  are  prime  numbers;  but  4,  6,  8,  9,  12, 
are  not  prime  numbers;  since  they  have  the  divisors,  2  and  3. 
Two  numbers  are  said  to  be  prime  with  respect  to  each  other, 
when  they  have  no  other  common  divisor  except  unity ;  thus, 
4  and  9,  7,  1,  and  12,  are  numbers  prime  with  respect  to  each 
other;  8  and  12  are  not,  since  they  are  divisible  at  the  same 
time  either  by  2  or  4. 

First  Principle.  —  Every  number,  which  exactly  divides  an- 
other number,  divides  also  any  multiple  whatever  of  this  second 
number. 

For  example,  24  being  divisible  by  8,  and  giving  for  a  quo- 
tient 3,  5  times  24,  or  120,  divided  by  8,  will  give  (No.  40)  for 
quotient,  5  times  3,  or  15.  In  the  same  manner,  60  being  divi- 
sible by  12,  and  giving  for  quotient  5,  7  times  60,  or  420,  divided 
by  12,  will  give  for  quotient  7  times  5,  or  35. 

Second  Principle.  —  Every  number,  decomposed  into  two 
parts,  both  divisible  by  a  second  number,  is  itself  divisible  by 


70  FRACTIONS. 

this  second  number.  For  the  quotient  of  the  division  of  the  total 
being  equal  to  the  sum  of  the  two  partial  quotients,  if  these  two 
partial  quotients  are  entire,  their  sum,  or  the  total  quotient,  must 
be  entire. 

Third  Principle.  —  Every  number  which  divides  exactly  a 
whole,  decomposed  into  two  parts,  and  which  divides  one  of  these 
parts,  ought  to  divide  also  the  other  part.  For  the  total  quotient 
being  equal  to  the  sum  of  the  two  partial  quotients,  if  one  of 
these  partial  quotients  is  fractional,  it  would  follow  that  an  entire 
number  would  be  equal  to  a  fractional  number ;  which  would  be 
absurd. 

51.  So  much  being  established,  let  the  two  numbers.,  360  and 
276,  be  given,  of  which  we  propose  to  determine  the  greatest  com- 
mon divisor,  or  the  greatest  number  which  will  divide  them  both 
exactly.  It  is  at  once  evident  that  this  greatest  common  divisor 
cannot  exceed  the  smaller  number,  276;  and  as  276  divides 
itself,  it  follows,  that  if  it  will  divide  360  also,  it  will  be  the 
greatest  common  divisor  sought.  Attempting  this  division  of 
360  by  276,  we  find  for  quotient,  1,  and  remainder,  84 ;  then, 
276  is  not  the  greatest  common  divisor.  We  say,  now,  that  the 
greatest  common  divisor  of  360  and  276,  is  the  same  as  that 
which  exists  between  the  smaller  number,  276,  and  the  remain- 
der of  the  division. 

For  the  greatest  common  divisor  sought,  since  it  ought  to 
divide  360,  and  one  of  its  parts,  276,  divides  necessarily  the 
other  part,  84,  (50) ;  whence,  we  can  conclude  at  once,  that  the 
greatest  common  divisor  of  360  and  276,  cannot  exceed  that 
which  exists  between  276  and  84 ;  since  it  must  divide  these  two 
numbers.  In  the  second  place,  the  Gr.  C.  D.  of  276  and  84, 
dividing  the  two  parts,  276  and  84,  of  360,  divides  necessarily 
this  number;  being  the  exact  divisor  of  360  and  276,  it  cannot 
exceed  the  greatest  C.  D.  of  360  and  276.  Whence,  we  see, 
that  the  G-.  C.  D.  of  360  and  276,  and  the  G.  C.  D.  of  276  and 
84,  cannot  be  greater  than  each  other;  then  they  are  equal. 


FRACTIONS.  71 

Thus,  tLe  question  is  reduced  to  seeking  the  greatest  C.  D.  of 
276  and  84 ;  numbers  simpler  than  360  and  276. 

"We  now  reason  on  276  and  84,  as  we  have  about  the  primitive 
numbers;  that  is,  we  try  the  division  of  276  by  84;  because, 
if  the  division  is  exact,  84  will  be  the  G.  C.  D.  of  276  and  84, 
and  consequently  of  360  and  276. 

EiFecting  this  new  division,  we  have  3  for  quotient,  and  24  for 
remainder;  then  84  is  not  the  Gr.  C.  D.  sought.  But,  by  analogous 
reasoning  to  that  above,  we  can  prove  that  the  Gr.  C.  D.  of  276 
and  84.  is  the  same  as  that  of  the  first  remainder,  84,  and  the 
second  remainder,  24.  The  question  is  then  reduced  to  finding 
the  G.  C.  D.  of  84  and  24;  we  divide  84  by  24,  and  obtain  3 
for  quotient,  and  12  for  remainder;  then  24  is  not  the  G.  C.  D. ; 
but,  as  this  G.  C.  D.  is  the  same  as  that  of  24  and  12,  we  divide 
24  by  12 ;  we  find  an  exact  quotient,  2 ;  thus,  12  is  the  greatest 
C.  D.  of  12  and  24,  hence  of  84  and  24,  of  276  and  84,  and, 
finally,  of  360  and  276,  or  the  G.  C.  D.  sought. 

In  practice,  we  arrange  the  operation  thus : 
13      3      2 


360 

84 


276 
24 


84 
12 


24 
0 


12 


After  having  divided  360  by  276,  we  place  the  quotient,  1, 
above  the  divisor,  and  a  remainder,  84 ;  we  write  this  remainder 
to  the  right  of  the  less  number,  276,  and  we  divide  276  by  84, 
placing  the  quotient,  3,  above  the  divisor,  and  the  remainder,  24, 
to  the  right  of  the  84,  and  so  on  for  the  rest. 

General  Rule.  —  In  order  to  find  the  G.  C.  D.  of  two  num- 
hers,  divide  the  greater  number  hy  the  less;  if  there  is  no  remain- 
der, the  smaller  number  is  the  G.  C.  D. 

If  there  is  a  remainder,  divide  the  less  number  by  this  remain- 
der }  and  if  this  division  is  without  remainder,  the  first  remain- 
der is  the  G.  0.  D. 

If  this  second  division  gives  a  remainder,  divide  the  first  re- 
mainder by  thf  i'wnd,  and  continu*   always  to  divide  the  pre- 


72 


FRACTIONS. 


ceding  remainder  hy  the  last  remainder,  until  the  division  becomes 
exact;  then  the  last  divisor  employed  will  he  the  G.  0.  D. 
sought. 

If  the  last  divisor  is  unity,  it  is  a  proof  that  the  two  numbers 
are  prime  with  respect  to  each  other.  Reciprocally,  if  two  num- 
bers are  prime  with  each  other,  and  if  we  apply  the  method 
above,  we  will  find  necessarily  a  final  remainder  equal  to  unity. 
For,  according  to  the  nature  of  the  method,  the  remainders  go 
on  diminishing ;  besides,  we  cannot  obtain  a  remainder,  nothing, 
before  having  obtained  a  remainder,  1 ;  since  the  divisor,  different 
from  unity,  which  gave  this  remainder  nothing,  would  be  the 
common  divisor  of  the  two  numbers.  Thus,  we  must,  necessa- 
rily, after  a  smaller  or  greater  number  of  operations,  obtain  unity 
for  a  remainder. 

52.  We  give  now  the  application  of  this  method. 
Reduce  the  fraction,  |||,  to  its  simplest  form. 

2 

18_5  I  /, 

00 

37 
16 

0  00 

We  find,  for  the  greatest  common  divisor,  37,  and,  dividing 
999,  and  592,  by  37,  we  have  ^f ,  for  the  value  of  |||,  reduced 
to  its  least  terms. 

If  we  can  find  no  common  divisor  greater  than  unity  for  the 
terms  of  the  fraction,  the  fraction  is  irreducible,  its  terms  being 
prime  with  each  other. 

Remark.  —  If,  in  the  operation  for  the  common  divisor,  we 
arrive  at  a  prime  number  for  a  remainder,  as,  for  example,  5  or 
7,  we  can  conclude  at  once  that  unless  this  prime  remainder 
exactly  divides  the  last  divisor,  the  two  primitive  numbers  have 
no  common  divisor  greater  than  unity.  The  reason  of  this  is 
obvious.     We  will  return  once  more  to  the  operation  of  the 


1 

1 

999 

592 

407 

407 

185 

37 

999 

37 

592 

259 

27 

222 

FRACTlOxNS.  73 

greatest  common  divisor,  as  it  is  one  of  the  most  important  in 
the  arithmetic. 

Second  example,  f  |§g|.  We  find  for  G-.  C.  D.  1261,  respective 
quotients,  29  and  23;  thus,  ||  is  the  fraction  reduced  to  its 
simplest  terms. 

53.  From  what  precedes,  it  results,  that  if  from  the  two  terms 
of  a  fraction,  we  subtract  the  same  multiples  of  the  two  terms 
of  the  equivalent  irreducible  fraction,  the  resulting  fraction  is 
also  equivalent  to  the  given  one. 

Let  us  take,  for  example,  the  fraction,  ^|,  which,  reduced  to 
its  least  terms,  according  to  the  method  indicated  in  the  preceding 
article,  is  equal  to  |.  If,  from  the  two  terms,  18  and  24,  of  the 
given  fraction,  we  subtract  four  times  3,  or  12,  and  four  times  4, 
or  16,  we  obtain  a  new  fraction,  |,  which,  expressed  in  simpler 
terms  than  those  of  the  given  fraction,  is  equal  to  it.  For,  in 
suppressing  the  factor,  2,  common  to  6  and  8,  we  find  |,  as  for 
the  first  fraction,  ||. 

It  is  easy  to  explain  this  result.  For,  if  ^|  is  equal  to  |,  an 
irreducible  fraction,  of  which  the  two  terms  are  prime  with  each 
other,  18  and  24  must  be  the  same  multiples  (6  times  3,  and  6 
times  4),  of  the  two  terms  of  the  fraction,  | ;  and  when  from  18 
and  24  we  subtract  four  times  3,  and  four  times  4,  we  obtain 
difi'erences,  twice  3,  and  twice  4,  which  are  also  the  same  multi- 
ples of  3  and  4;  whence  results  a  new  fraction,  |,  equal  to  |. 
It  would  seem  that  this  proposition  ought  to  furnish  a  means  of 
simplifying  a  fraction ;  but  we  see  that  this  means  would  be  alto- 
gether illusory,  since  it  supposes  the  irreducible  fraction  known, 
to  which  the  given  one  is  equivalent. 

N.  B.  We  would  remark  here,  that  we  subtract  from  the  two 
terras  of  the  fraction  two  different  numbers,  and  not  the  same 
number  as  in  (48). 

We  pass,  now,  to  the  four  fundamental  operations  upon  frac- 
tions. 

7 


74  FKACTIONS. 


Addition  of"  Fractions. 

64.  The  addition  of  fractions  has  for  its  object  to  find  a  single 
fraction  which  shall  express  the  value  of  the  sum  of  several 
fractions. 

There  are  two  cases ;  the  fractions  to  be  added  are  either  of 
the  same  species j  that  is  to  say,  have  the  same  denominator ^  or 
of  different  species. 

In  the  first  case,  we  sutti  up  the  numerators^  and  then  give  to 
this  sian  the  common  denominator. 

In  the  second  case,  we  reduce  the  fractions  to  the  same  deno- 
minator ;  after  ichich  we  operate  as  in  the  first  case.  The  reason 
is  obvious,  since  the  denominator  is  a  sign  indicating  the  value 
or  species  of  the  units  to  be  added,  and  the  numerator  the  num- 
ber of  these  units. 

^.         2  ^3  ^4      2+3+4     9 

Thus,  n+n+n=    iir-==iT 

In  the  same  manner, 

5_  .  1  ,  1  ,  i  _5+2+7+4_18 
23  "^23  "'■23  ■^23""        23         "~23 

Let  it  be  given,  now,  to  add  the  three  fractions, 

2  3  7 

3  4  H 

8  6  3 

16  18  21 

54  24  24 

After  having  reduced  these  fractions  to  the  least  common  de- 
nominator, 24,  (No.  46),  we  add  the  numerators,  16,  18,  and 
21 ',  we  then  give  to  the  sum,  55,  the  denominator,  24. 

We  have  thus  ||  for  the  result  required. 

55.  This  last  example  ISads  to  a  fractional  expression,  ^|, 
greater  than  unity,  which  gives  rise  to  a  new  operation. 

We  have  seen  that  unity  is  equivalent  to  ||,  or  twenty-four 
twenty-fourths;  whence,  it  follows,  that  as  many  times  as  55 
contains  24,  so  many  units  there  are  in  4|.     Now,  dividing  55 


FRACTIONS.  75 

by  24,  we  have  for  a  quotient,  2,  with  remainder,  7 ;  thus,  j-^  is 
a  number  composed  of  2  units  and  ■^^.  In  general,  when  we 
obtain  a  fractional  result,  of  which  the  numerator  exceeds  the 
denominator,  in  order  to  extract  the  whole  number  contained  in 
this  expression,  ice  must  divide  the  numerator  hy  the  denomina- 
tor. The  quotient  lohich  loe  obtain  represents  the  entire  number ^ 
and  the  remainder  is  the  numerator  of  the  fraction  which  is  to 
be  added  to  the  entire  number,  (43). 
By  this  mode,  we  find, 

17 1    5    .     153 10  3  ini  •     65  4 — 73  1 

Reciprocally,  when  we  have  an  entire  number  joined  to  a  frac- 
tion, in  order  to  form  a  single  fractional  number,  we  must  mul- 
tiply the  entire  number  by  the  denominator,  add  the  product  to  the 
numerator,  and  (jive  to  the  sum  the  denominator  of  the  fraction. 

For  example, 

_2     3x5  +  2     17     ,,  ^      11x12+7     139     „ 

Subtraction  of  Fractions. 

66.  The  subtraction  of  fractions  has  for  its  object  to  find  the 
excess  of  a  greater  fraction  over  a  smaller. 

If  the  two  fractions  have  the  same  denominator,  we  subtract 
the  smaller  numerator  from  the  greater,  and  give  to  the  difference 
the  common  denominator. 

If  they  have  not  the  same  denominator,  we  reduce  them  to 
such  as  have ;  after  which,  we  proceed  as  in  the  first  case. 

Given,  to  subtract  -^^  from  |i ;  there  remain  y^^,  or  1.  In 
the  same  manner,  ^|  —  ^'^^  =  1  J=-J'^. 

Given,  to  subtract  |  from  |.  These  two  fractions  give  ^|  and 
§i)  by  reduction  to  the  same  denominator;  and  we  have 

21      16_21— 16      5 
24      24  ~"     24     ~24* 

19      13_19x  17  — 13x20      63 
'  20      17~         20x17         ""340' 


76  FRACTIONS. 

We  can  have  an  entire  number  joined  to  a  fraction,  to  be  sub- 
tracted from  an  entire  number  joined  to  a  fractiom ;  or,  as  they 
are  called,  a  mixed  numher,  to  be  subtracted  from  a  mixed 
number. 

Griven,  for  example,  to  subtract  b\l  from  12|. 
12|=12f|=ll|< 
5H=  5J|=  5JI 


6i|. 

We  commence  by  reducing  the  two  fractions  to  the  same  de- 
nominators, which  gives  ||  for  the  first,  and  ||  for  the  second. 

Then,  as  we  cannot  subtract  |4  from  |J,  we  take  from  the 
entire  part,  12  of  the  greater  number,  one  unit,  which  we  add 
to  the  II,  making  |i )  we  then  subtract  ||  from  |^,  and  have 
for  a  remainder,  4^.  Passing  to  the  subtraction  of  the  entire 
numbers,  we  regard  the  greater  number  as  diminished  by  unity, 
and  subtract  5  from  11,  which  gives  6.  We  have  thus  6||  for 
the  required  result. 

The  same  result  could  be  obtained  by  reducing  the  mixed 
numbers  to  single  fractions  by  last  article,  and  then  following  the 
rule  given  for  subtraction  of  fractions. 

Multiplication  of  Fractions. 

57.  Multiplication  has  for  its  object  in  general,  two  numher s 
being  given  to  form  a  third  number,  which  is  compounded  with 
the  first  number  in  the  same  manner  as  the  second  number  is 
compounded  with  unity. 

This  being  established,  we  distinguish  three  principal  cases  in 
the  multiplication  of  fractions.     We  can  have, 

1st.   A  fraction  to  be  multiplied,  hy  an  entire  number. 

Given,  for  example,  ^^^  to  be  multiplied  by  5. 

According  to  the  definition  above,  since  the  multiplier,  5,  con- 
tains 5  times  unity,  it  follows,  that  the  product  ought  to  be  equal 
to  5  times  y^^,  or  5  times  as  great  as  -^^.     Now,  we  have  seen  in 


FRACTIONS.  77 

(43),  that  we  render  a  fraction  5  times  greater  by  multiplying  its 

numerator  by  5.  We  thus  have =^ or  — ,  for  the  required 

product. 

Then,  in  order  to  multipli/  a  fraction  hy  an  entire  number,  we 
must  m,ultiply  the  numerator  hy  the  entire  number,  and  give  to 
the  product  the  denominator  of  the  fraction. 

Given,  to  multiply  jg  by  9. 

We  obtain  f  |  for  the  product,  or  5y®g,  or  5^.  This  result  can 
be  obtained  more  simply  thus.  For,  by  (43),  we  can  divide  the 
denominator  by  9,  instead  of  multiplying  the  numerator.  And 
we  find  thus,  y ,  or  51,  for  the  required  product. 

We  can  only  apply  this  last  method,  when  the  denominator  is 
divisible  by  the  number.  The  established  rule  is  always  appli- 
cable. Usage  alone  renders  us  familiar  with  these  simplifications. 

2d.    To  multiply  an  entire  number  by  a  fraction. 

Example.  — 12  to  be  multiplied  by  ^. 

Since,  in  this  case,  the  multiplier,  ^,  is  equal  to  4  times  the 
7th  part  of  unity,  the  product  ought  to  be  equal  to  4  times  the 
7th  of  12.  Now,  the  7th  of  12  is  equal  to  ^^  ]  and,  in  order  to 
render  this  4  times  as  great  as  L^^  -^^e  must  multiply  the  nume- 
rator by  4 ;  we  thus  obtain  %f,  or  6f ,  for  the  required  product. 

Then,  to  multiply  an  entire  number  by  a  fraction,  we  multiply 
the  entire  number  by  the  numerator,  and  give  to  the  prodtict  the 
denominator  of  the  fraction. 

Thus,  29  X  ^=:i^  =  25|. 

We  might  find  this  last  result  by  dividing  24  by  6,  and  multi- 
plying the  result  by  5. 

But,  we  repeat,  these  simplifications  are  not  always  possible. 

7=^ 


78  FRACTIONS. 

3d.  A  fraction  to  be  multiplied  hy  a  fraction. 

Example.  —  Griven,  to  multiply  |  by  |. 

The  reasoning  is  analogous  to  that  of  the  preceding  case; 
since  I  is  equal  to  5  times  the  8th  part  of  unity,  the  product 
ought  itself  to  be  5  times  the  8th  part  of  the  multiplicand,  |. 
Now,  in  order  to  take  the  8th  of  |,  we  must  (43)  multiply  the 
denominator  by  8,  which  gives  Z^-;  and  in  order  to  obtain  a 
fraction  5  times  as  great  as  ^^,  we  must  multiply  the  numerator 
by  5;  which  gives  ^j  for  the  product  required. 

Then,  to  multiply  one  fraction  hy  another,  multiply  numerator 
hy  numerator  J  and  denominator  hy  denominator  ;  then  make  the 
second  product  denominator  of  the  first. 

We  find,  thus, 

12^    6  ~  72* 

,    ^    8        3        24       2 

^^^r5^T  =  6o  =  y- 

N.  B.  In  the  two  preceding  cases,  the  product  is  always  less 
than  the  multiplicand ;  and  this  ought  to  be  the  case,  since  the 
operation  is  really  taking  a  part  of  the  multiplicand  indicated  by 
the  fractional  multiplier. 

58.  Finally,  one  of  the  factors  of  the  multiplication,  or  both 
of  them,  may  be  mixed  numbers.  These  numbers  are  equivalent, 
respectively,  to  the  improper  fractions,  (the  fractions  greater 
than  unity  being  called  improper  fractions),  ^^  and  "^^ ;  per- 
forming the  multiDlication  of  these  by  the  rule  above,  we  obtain 
^§fSor453V. 

We  could  effect  this  multiplication  by  parts ;  that  is  to  say, 
multiply,  first,  7  by  5,  |  by  5,  7  by  |,  and  |  by  | ;  then  add  these 
four  products ;  but  this  method  is  much  the  longest. 


FRACTIONS.  79 


Division  op  Fractions. 

59.  Division  has  for  its  object:  Given.y  the  product  of  two 
factors,  and  one  of  the  factors  to  determine  the  other. 

It  results,  obviously,  from  this  definition,  and  from  that  of 
multiplication,  that  the  first  number,  called  dividend,  is  com- 
pounded with  the  third,  called  quotient,  in  the  same  manner  that 
the  divisor  is  compounded  with  unity. 

This  established,  in  the  division  as  in  the  multiplication  of 
fractions,  we  distinguish  three  principal  cases. 

1st.    To  divide  a  fraction  hy  an  entire  number. 

Given,  for  example,  |,  to  be  divided  by  6.     Since  the  divisor 

is  6  times  unity,  it  follows,  that  the  dividend,  f ,  is  equal  to  6 

times  the  required  quotient;   then,   reciprocally,   the  quotient 

ought  to  be  the  6th  part  of  f .     Now,  in  order  to  take  the  6th 

part  of  a  fraction,  or  to  obtain  one  6  times  as  small,  we  must  (43) 

5  5 

multiply  the  denominator  by  6 ;  thus,  we  obtain  — -: -,  or  t-^, 

for  the  required  quotient. 

Then,  to  divide  a  fraction  by  an  entire  number,  multiply  the 
denominator  of  the  fraction  hy  tlie  entire  number,  leaving  the 
numerator  the  same. 

Thus,  11  divided  by  8  =  -^  =  |1. 

23  23 

In  the  same  manner,  ^  divided  by  12  =  -^^. 
oU  360 

The  quotient  of  ^f  by  6,  is  -^f^ ;  but  we  can  efiiect  the  divi- 
sion of  Jl  by  6,  by  taking  the  6th  of  the  numerator,  which  gives 
■^j^;  the  same  with  -^f^,  when  we  suppress  the  factor,  6,  com- 
mon to  the  two  terms.  Then  we  add  to  the  above  rule,  or  divide 
the  numerator  hy  the  divisor,  when  that  is  possible. 

2d.    To  divide  an  entire  number  by  a  fraction. 

Given,  to  divide  12  by  J. 


80  FRACTIONS. 

Since  the  divisor,  |,  is  equal  to  7  times  the  9th  part  of  unity, 

it  follows,  that  the  dividend  is  also  equal  to  7  times  the  9th  part 

of  the  required  quotient.     Then,  taking  the  7th  of  12,  which 

gives  L2^  we  will  have  the  9th  of  the  quotient  sought;  and  to 

obtain  this  quotient  itself,  we  must  take  9  times   ^^f,  which  is 

done   by   multiplying   the   numerator   by   9;    we   thus   obtain 

9  times  12         108  ,,-,., 
,  or  — ,  equal  to  15|. 

Then,  in  order  to  divide  an  entire  number  hy  a  fraction,  we 
must  multiply  the  entire  7iumher  hy  the  denominator,  and  divide 
the  product  hy  the  numerator. 

Or,  we  can  say,  as  we  have  here  multiplied  12  by  |,  multiply 
the  entire  numhcr  hy  the  fraction  inverted. 

3d.    To  divide  a  fraction  hy  a  fraction. 

Griven,  to  divide  |  by  -f^. 

The  reasoning  is  like  the  preceding.  The  divisor,  -^j,  being  8 
times  the  11th  part  of  unity,  the  dividend,  |,  is  also  equal  to  8 
times  the  11th  of  the  quotient;  then,  the  8th  of  |,  or  -^^^  is  the 
11th  of  the  quotient;  and  11  times  -f^,  or  ||,  is  the  quotient 
sought. 

Then,  to  divide  a  fraction  hy  a  fraction^  we  must  multiply  the 
numerator  of  the  dividend  hy  the  denominator  of  the  divisor, 
and  the  denominator  of  the  dividend  hy  the  numerator  of  the 
divisor ;  then  make  the  second,  product  the  denominator  of  the 
first. 

Or,  in  simple  terms,  multiply  the  dividend  hy  the  divisor  vnth 
its  terms  inverted. 


Thus,  1-^ 

•f  = 

1  times 

7 2  1 1 

ft — 30  —  - 

L3V 

In  the 

same  manner. 

28  .  13 

30  •  r5~ 

23 

'30 

15 

^r3  = 

23  X  15 
"30x13 

345 
~390' 

(We  could  have  suppressed  the  factor,  15,  obviously  common 
to  both  terms  of  the  product  in  this  last  example,  before  perform- 
ing the  multiplication). 


FRACTIONS.  81 

N.  B.  Whenever,  in  the  division  of  fractions,  the  divisor  is 
less  than  unity,  the  quotient  will  be  greater  than  the  dividend. 
For  this  quotient  results  from  the  multiplication  of  the  dividend 
by  the  divisor  inverted,  a  number  greater  than  unity. 

60.  Finally,  if  we  have  a  mixed  number,  we  reduce  first  the 
entire  parts  to  fractions,  and  then  proceed  as  in  the  case  above. 

Given,  12|  to  be  divided  by  6|.     We  have 

193_:_fi2 51_i_20— 51v     3    153 

In  the  same  manner, 

7  8  _i_1  Pis 85_i.l25  —  85  V      8    680 

Remarh. — The  rules  for  the  division  and  multiplication  of 
fractions  can  be  very  readily  deduced  by  regarding  them  as  un- 
performed divisions. 

Remarh  II.  —  It  is  evident  that  the  division  of  fractions  can 
give  rise  to  fractions  with  fractional  terms,  or  complex  fractions^ 
as  they  are  sometimes  called.     We  can  have,  for  example, 

1,   ^j:!!A,   I  ^^^^^  i^ 

f        24  +  1       4f  times  f 

Which  are  reduced  to  fractions  of  two  terms  by  performing  all 
the  operations  indicated  upon  the  separate  fractions,  according  to 
known  rules. 

Fractions  op  Fractions. 

61.  To  the  multiplication  of  fractions  attaches  itself  another 
species  of  operation,  known  under  the  name  of  the  ride  for  frac- 
tions of  fractions,  or  compound  fractions. 

In  order  to  give  an  accurate  idea  of  this  operation,  suppose, 
first,  that  we  have  to  take  a  part  of  the  fraction,  |,  indicated  by 
the  fraction,  |,  As  this  is  the  same  thing  as  taking  twice  the 
third  of  f,  or  (57),  multiplying  |  by  |,  we  have  for  result, 

5  times  2        10 
7  times  3'       21* 


82  FRACTIONS. 

Suppose,  DOW,  we  wish  to  take  a  part  of  JJ,  indicated  by  the 
fraction,  -f.^  ;  we  would  have,  as  above,  — — — -  =-——,  and  this 
last  expression  would  represent  -^^  of  |  of  |. 

Hence,  we  see,  that  in  order  to  take  fractions  of  fractions,  we 
must  multiply  all  the  numerators  together j  and  all  the  denomi- 
nators together,  and  give  the  last  product  as  denominator  to 
the  first. 

When  we  have  to  take  fractions  of  fractions  of  a  given  entire 
number,  we  put  this  entire  number  under  the  form  of  a  fraction, 
having  1  for  denominator,  and  apply  the  rule  which  has  just  been 
established. 

Thus,  the  |  of  |  of  f  of  f  of  \^=  ^  •  3  •  | ;  «  ;  ^^^  =  M^,  or, 
reducing,  3^ 44  ^  3_3_. 

We  can  simplify  these,  and  similar  operations,  by  suppressing 
the  factors  common  to  both  terms. 

Thus,  in  the  example,  |  of  |  of  -{^^  of  ^  of  \^,  if  we  suppress 
the  common  factors,  we  have 

3x5 


or  y,  or  7^. 


Approximative  Valuation  of  Vulgar  Fractions. 

62.  Tn  order  to  complete  the  general  theory  of  fractions,  we 
will  resolve  the  following  question,  which  has  many  useful  appli- 
cations. 

Given,  an  irredueihle  fraction,  of  luhich  the  terms  are  so  large, 
that  it  is  dijfficuH  to  form  an  accurate  idea  of  its  value,  to  replace 
it  hy  another  ichich  approaches  it  in  value  to  within  certain 
limits,  but  whose  terms  are  much  more  simple;  that  is,  which 
have  for  denominators,  2,  3,  4,  5,  6,  &c. 

Take,  for  example,  the  fraction,  |||.  We  propose  to  find  its 
approximate  value  in  twelfths,  (i.  e.)  to  replace  it  by  a  fraction 
having  12  for  denominator. 


FRACTIONS.  83 

We  remark,  first,  that  unity  being  equal  to  if,  |f|  of  unity 

523  X  12 
are  equal  to  f  f  j  of  j|,  or  equal  to  g^g^^-     Multiplying    523 

by  12,  we  obtain  6276 ;  which,  divided  by  949,  gives  6  for  quo- 
tient, and  582  for  remainder.    Then,  the  fraction  is  y^^,  with  the 

582 
remainder,  niQ-^o,  less  than  j\.     Hence,  -j^^  is  the  value  of 

the  fraction  to  within  less  than  j^^. 

63.  In  general,  in  order  to  transform  a  fraction,  -,  into  ano- 
ther having  a  denominator,  n,  at  the  same  time  differing  from  the 
first  by  less  than    - ,  we  have  the  following  rule. 

Multiply  the  numerator  of  the  proposed  fraction  hy  7i,  and 
divide  the  product  hy  the  denominator. 

Form  J  then,  a  new  fraction,  having  for  numerator  the  entire 
part  of  the  quotientj  and  n  for  denominator. 

General  Observations  on  Fractions. 

64.  It  results,  obviously,  from  the  nature  of  the  methods 
established  for  the  calculus  of  fractions,  that  the  four  fundamen- 
tal operations  performed  upon  them,  to  wit :  addition,  subtrac- 
tion, multiplication,  and  division,  are  reduced  always  to  the  same 
operations  performed  on  entire  numbers. 

Thus,  for  example,  the  addition  and  subtraction  of  fractions 
is  brought  back,  by  the  reduction  to  common  denominators,  to 
the  addition  and  subtraction  of  their  numerators. 

In  the  same  manner,  multiplication  of  fractions  is  effected  by 
multiplying  the  numerators  together,  and  the  denominators. 
Division  of  fractions  becomes  multiplication,  after  inverting  the 
divisor. 

We  conclude  from  this,  that  the  principles  established  in 
Nos.  25  and  26,  upon  the  multiplication  of  entire  numbers, 
are  equally  applicable  to  fractions ;  that  is  to  say,  1st,  to  multiply 
a  fraction  by  the  product  of  several  others,  is  the  same  thing  as 


84  FRACTIONS. 

to  multiply  the  first  fraction  successively  hy  each  one  of  the  factors 
of  the  product:  2d,  the  product  of  two  or  more  fractions  is  the 
same  in  whatever  order  we  perform  their  multiplication.  In 
fine,  we  can  apply  to  fractions  all  the  propositions  established  in 
(40),  concerning  the  changes  which  the  product  of  a  multiplica- 
tion, or  the  quotient  of  a  division  undergo,  when  we  cause  one 
of  the  terms  of  the  operation  to  undergo  certain  changes.  We 
can  multiply  or  divide  both  terms  of  any  fractional  expression 
whatever  by  the  same  number,  without  altering  its  value;  and 
so  on  of  other  principles.  We  can  deduce  from  the  definition 
of  multiple  and  submultiple,  or  divisor  of  a  number,  that  there 
exist  fractions  which  are  multiples  and  submultiples  of  other 
fractions,  in  the  sense  that  the  division  of  the  multiple  fraction 
by  the  submultiple  gives  an  entire  quotient.  Thus,  the  fractions, 
J|,  3j^3,  2^3,  •  •  •  •  are  multiples  of  3^^,  since  they  contain  the 
latter,  6,  4,  3  ...  .  times,  without  remainder. 

In  general,  every  fraction  has  for  divisors  its  half,  its  third,  its 
fourth,  &c. ;  whence,  it  follows,  that  the  number  of  its  divisors 
is  infinite,  which  is  not  true  of  entire  numbers,  if  the  divisors 
are  to  be  entire. 

Two  fractions  can  also  have  common  divisors;  thus,  ||,  -^^^ 
have  for  common  divisor  the  fraction,  -^^^  and  all  its  submulti- 
ples;  for  the  quotients  of  ||  and  -^^j  divided  by  -^^^  are  re- 
spectively 5  and  2,  entire  numbers.  We  can,  then,  generally 
establish,  with  relation  to  fractions,  properties  analogous  to  those 
which  we  have  proved  concerning  the  greatest  common  divisor, 
and  the  least  common  multiple  of  two  or  more  numbers. 


Of  Tvji 


85 


CHAPTER  III. 

COMPOUND  NUMBERS. 

65.  The  theory  of  compound  numbers  we  place  here  as  an  im- 
mediate application  of  the  theory  of  vulgar  fractions.  The  units 
of  smaller  denominations  being  fractions  of  the  principal  units, 
or  units  of  higher  denominations,  and  fractions  being  really  no- 
thing more  than  units  of  lesser  value  than  the  principal  unit  with 
which  they  are  compared.  We  can  thus,  in  the  number,  5|, 
regard  the  ninths  as  simple  units,  and  5  as  a  number  made  up 
of  compound  units,  each  one  equal  to  9  times  the  simple  unit  3 
and  the  9  under  the  4  is  the  sign  or  denominator,  showing  the 
relative  value  of  the  simple  units  expressed  by  the  number,  4. 
Thus,  we  have  seen,  in  (No.  8),  that  in  order  to  value  quantities 
smaller  than  the  principal  unit,  we  conceive  this  unit  divided 
into  a  certain  number  of  equal  parts,  which  we  regard  as  forming 
new  units.  In  the  theory  which  now  occupies  our  attention,  the 
principal  unit  is  first  divided  into  a  small  number  of  equal  parts, 
then  these  are  divided  into  others,  and  these  new  parts  into 
others,  &c.,  &c. 

Thus,  for  coin,  the  pound  sterling,  English,  is  divided  into  20 
parts,  called  shillings ;  the  shillings  into  12  parts,  called  pence, 
&c.  In  the  same  manner,  the  unit  of  length,  the  yard,  is  divided 
into  3  parts,  called  feet ;  the  foot  into  12  parts,  called  inches,  &c. 
66.  Every  art,  each  trade,  each  country,  subdivides  the  prin- 
cipal unit,  according  to  its  own  method. 

The  following  tables  give  for  the  most  important  o^ these  quan- 
tities, the  principal  units  and  their  subdivisions ;  that  is  to  say, 
those  which  follow  the  analogies  of  vulgar  fractions.  The  deci- 
mal divisions  of  the  principal  units  we  reserve  for  the  chapter  on 
decimal  fractions. 
8 


86  COMPOUND   NUMBERS. 

TABLES 

In  the  estimation  of  time,  the  year  is  adopted  as  the  principal 
unit  j  the  subdivisions  being,  months,  weeks,  days,  hours,  minutes, 
peconds. 

The  year  is  divided  into  365  days. 

The  day  24  hours. 

The  hour  60  minutes. 

The  minute 60  seconds. 

(The  minutes  and  seconds  are  generally  indicated  by  '  and  ".) 
Or  we  may  write  the  table  thus : 

One  second  =  ^^  of  a  minute. 
One  minute  =  g*^  of  an  hour. 
One  hour      =  ^^  of  a  day. 
One  day        =  ^  of  a  week. 

Ex.  —  5  days,  6  hours,  25  minutes,  and  36  seconds,  may  be 
written  either  in  columns,  bd,  6h,  25',  36",  or  thus : 

6  +  ^4  +  U  of  5\  +  IS  of  5*0  of  3S\- 

COINS. 

Of  coins,  we  give  only  the  chief  divisions  of  the  English  cur- 
rency; the  American  and  French  coming  under  the  decimal 
systems. 

English  Money. 

One  pound  sterling  =  £  is  divided  into  20  shillings. 

One  shilling  =s 12  pence. 

One  penny  =  c? 4  farthings. 

Or  we  may  write  it  thus  : 

One  farthing  =  ^J  of  a  penny. 
One  penny      =  y^^j  of  a  shilling. 
One  shilling  =  .^^  of  a  pound. 

Ex.  —  5  pounds  sterling,  6  shillings,  and  10  pence,  may  bo 
written  £5,  6s.,  10^.,  or  £5  -\-^  +  |?  of  -^-q. 


COMPOUND    NUMBERS.  87 


WEIGHTS. 


The  standard  avoirdupois  pound  of  the  United  States,  is  the 
weight  of  27-7015  cubic  inches  of  distilled  water  weighed  in 
air,  at  a  fixed  temperature.  This  gives  us  a  fixed  unit  of  com- 
parison, or  a  principal  unit  of  weight,  of  which  the  other  divi- 
sions of  the  table  are  either  multiples  or  submultiples. 


TABLE    OF  AVOIRDUPOIS   WEIGHT. 

The  ton is  divided  into  20  hundreds  =  cwt. 

The  hundred  weight 4  quarters    =  qrs. 

The  quarter  28  pounds      =  lbs. 

The  pound 16  ounces      =  oz. 

The  ounce 16  drams       =  dr. 

The  cwt.  in  this  table  contains  112  Ihs.,  but  the  cwt.  of  one 
hundred  pounds  is  very  generally  adopted  in  commerce,  as  more 
convenient,  and  much  better  adapted  to  the  decimal  system  of 
the  Federal  money. 

TROY   WEIGHT. 

The  standard  Troy  pound  of  the  United  States  is  the  weight 
of  22-794377  cubic  inches  of  distilled  water,  weighed  in  air  at 
a  given  temperature. 

,  TABLE. 

The  pound  (flb)  is  divided  into  12  ounces  =  oz. 

The  ounce 20  pennyweights  =  dwt. 

The  pennyweight  24  grains  =  grs. 

(7000  grains  Troy  make  1  lb.  avoirdupois.) 

The  Apothecaries'  weight  for  mixing  medicines  has  the  same 
principal  unit  as  the  Troy  weight,  but  difiers  only  in  its  subdivi- 
sions. 


COMPOUND   NUMBERS. 


TABLE. 


The  pound  (lb)  is  divided  into  12  ounces    =  ^ . 

The  ounce 8  drams    =5. 

Thedram  3scruples  =  9. 

The  scruple 20  grains    =r  gr. 

(The  English  pound,  Avoirdupois  and  Troy,  differ  a  little  from 
those  of  the  United  States). 

MEASURES    OF   LENGTH,   AREA   AND   VOLUME. 

Long  Measure. 

The  principal  unit  of  length  is  the  yard,  which  is  determined 
on  the  principle  in  physics  that  the  pendulum  which  vibrates 
once  in  a  second  at  the  same  place  on  the  earth's  surface,  under 
the  same  surrounding  circumstances,  has  a  fixed  and  invariable 
length.  This  pendulum,  or  metal  rod,  is  then  divided  off  accu- 
rately, and  a  certain  number  of  these  subdivisions  is  called  a 
yard.  For  the  United  States,  the  length  of  the  pendulum  is 
determined  in  New  York  city. 

TABLE. 

12  inches  make 1  foot. 

3  feet 1  yard. 

6  feet 1  fathom. 

5^  yards 1  pole  or  perch. 

40  poles 1  furlong. 

8  furlongs 1  mile. 

3  miles  1  league. 

MEASURE  OP  AREA,  OR  SQUARE  MEASURE. 

The  principal  unit  here,  with  which  surfaces  are  compared,  is 
a  square  whose  side  is  1  yard,  or  square  yard. 


COMPOUND    NUMBERS.  89 

TABLE. 

144  square  inches  make  =  1  sq.  foot. 

9  sq.  feet =  1  sq.  yard. 

30|  sq.  yards =  1  sq.  pole  or  percA. 

40  perches  =  1  rood. 

4  roods  =  1  acre. 

The  acre  then  contains  4840  sq.  yards.  For  larger  areas,  we 
have  the  square,  one  of  whose  sides  is  a  mile.  This  square  mile 
contains  640  acres,  (called  a  section  in  the  public  lands  of  the 
United  States). 

CUBIC,    OR   SOLID    MEASURE. 

The  unit  of  volume,  or  solid  measure,  is  a  cube  having  one 
yard  for  its  side,  the  other  divisions  being  either  multiples  or 
subdivisions  of  this. 

TABLE. 

1728  cubic  inches  =  1  cubic  foot. 

27  cubic  feet      =  1  cubic  yard,  &c.,  &c. 

The  relations  between  the  three  tables  of  long  measure,  square 
and  cubic  measure,  depend  upon  simple  geometrical  principles, 
which  the  student  will  find  developed  in  any  elementary  work 
upon  that  subject. 

LIQUID    MEASURE. 

The  standard  gallon  of  the  United  States  is  the  wine  gallon^ 
which  is  equal  to  231  cubic  inches. 

The  gallon  is  divided  into  4  quarts. 

The  quart 2  pints. 

The  pint 4  gills. 

For  the  higher  measures, 

63  gallons       =  1  hogshead. 
2  hogsheads  =^  1  pipe  or  butt. 
4  hogsheads  =  1  tun. 
8* 


90  COMPOUND    NUMBERS. 


DRY    MEASURE. 

The  principal  unit  is  the  bushel.  The  standard  bushel  of  the 
United  States  measures  21504  cubic  inches.  The  names  of  the 
subdivisions,  though  the  same  as  in  liquid  measure,  do  not  repre- 
sent the  same  volumes.  The  gallons^  quarts,  and  pints,  in  liquid 
measure,  measure  respectively,  231,  57|,  and  28|  cubic  inches; 
while  in  dri/  measure,  they  measure  268|,  67 J,  and  33|  cubic 
inches  respectively. 

TABLE. 

The  bushel  is  divided  into  4  pecks. 

The  peck  2  gallons. 

The  gallon  4  quarts. 

The  quart  2  pints. 

(The  English  imperial  gallon  measures  277*274  cubic  inches.) 

We  see  from  these  tables  the  great  importance  of  determining 
accurately  the  standard  of  length,  as  all  the  other  principal  units 
of  commerce  depend  upon  this.  Thus,  the  standard  of  dry  and 
liquid  measure  is  a  certain  number  of  cubic  inches.  The  standard 
weight  is  a  certain  number  of  cubic  inches  of  water.  The  standard 
of  money  is  a  coin  containing  a  given  weight  of  metal. 

67.  We  call  a  compound  number  every  concrete  or  denominate 
number,  which  contains,  at  the  same  time,  one  or  more  principal 
units  of  a  certain  species,  and  one  or  more  subdivisions  of  this 
unit,  or  simply  one  or  more  subdivisions  of  the  principal  unit 
alone.  Thus,  £10  12s.  Sd.,  2b  mis.  4:  fur.  1yds.,  70  days,  23 
hours,  10  min.,  or  simply  12s.  lOc/.,  4  A.  iOmin.,  &c.,  &c.,  are 
compound  numbers. 

But,  £10,  or  10s.,  or  23  hours,  are  not  compound  numbers, 
considered  thus  isolated.  The  resolution  of  the  following  ques- 
tions serves  as  a  basis  for  the  four  fundamental  operations  on 
compound  numbers. 


COMPOUND   NUMBERS.  91 

68.  Question  first.  —  A  compound  number  being  given^  to  re- 
duce this  number^  or  express  it  in  units  of  the  smallest  subdivi- 
sion of  the  principal  unit. 

Given,  for  example,  2lb.  4toz.  17 dwfs.  bgrs.,  to  be  converted 
into  grains. 

It  results  from  the  tables,  that  the  pound  equals  12  ounces. 

Therefore,  2 lb.  4 oz..=  2  X  12  +  4  =  28  02.,  or,  2/^ lb  =  f  |  lb. 
In  the  same  manner,  the  ounce  equals  20  dwt.  Hence,  28  oz., 
17  dwt.  =  28  X  20  +  17  pennyweights  =  577  dwt.,  or,  281^  = 
^y  oz.  Again,  577  dwt.  bgrs.  =  577  X  24  +  ^grs.  =  13853 
grains,  or,  5773j\  =  ^  \%^  ^  dwt. 

GrENERAL  RuLE.  —  Multiply j  first,  the  number  of  principal 
units  by  the  number  of  units  of  the  first  subdivision  which  the 
principal  unit  contains,  and  add  to  the  product  the  units  of  this 
first  division,  which  are  contained  in  the  given  number.  Then 
multiply  the  result  thus  obtained  by  the  number  of  units  of  the 
second  subdivision  which  the  first  contains,  and  add  to  this  second 
product  the  units  of  the  second  subdivision,  which  enter  into  the 
given  compound  number ;  and  thus,  in  succession,  until  we  arrive 
at  the  last  subdivision  or  denomination. 

We  will  find,  by  this  method, 

1st.  —  59  ft).  13  dwts.  5  gr.  =  340157  gr. 

2d.  — 121  lb.  Os.  d^d.         =  58099  halfpence. 

3d.  —  23  h.  55  min.  19''     =  26119  seconds. 

69.  Second  question.  —  Reciprocally,  given  a  number  of  units 
of  a  certain  division  of  the  principal  unit,  to  be  converted  into  a 
compound  number.  The  rule  to  be  followed  is  evident  from  what 
precedes,  and  can  be  enunciated  thus : 

Divide,  first,  the  proposed  number,  by  the  number  which  ex- 
presses how  many  times  the  given  subdivision  is  contained  in  the 
subdivision  next  higher ;  v:e  obtain  thus  for  quotient,  a  certain 
number  of  units  of  this  next  higher  division,  and  for  remainder 
the  units  of  the  given  denomination  which  are  to  enter  into  the 
compound  number  sought.      Divide,  then,  the  quotient  obtained 


92 


COMPOUND   NUMBERS. 


hy  the  numher  whicJi  expresses  liow  many  times  the  suhdivision 
next  higher  is  contained  in  the  denomination  higher  hy  txoo  than 
the  given  one  ;  we  obtain  a  new  quotient^  which  contains  a  certain 
numher  of  units  of  the  third  denomination^  of  which  we  have 
just  spoken,  and  a  new  remainder,  expressing  the  units  of  the 
denomination  next  to  the  given  one,  which  make  part  of  the  com- 
pound numher  sought.  Continue  thus,  until  the  quotients  cease 
to  he  divisible  by  the  numher  expressing  the  relation  between  the 
value  of  two  successive  denominations. 

N.  B.  If  we  obtain  0  for  any  one  of  the  remainders,  this  proves 
that  the  denomination  corresponding  is  wanting  in  the  number 
sought. 

Let  us  apply  this  rule  to  the  first  example  of  (68). 


13853  1 

190 

24 

20 

185 
173 

577 

177 
17 

28  1  12 

4   12 

Or  thus : 


13853  ^r. 
577  dwt. 
28  oz. 


Result,  21b.  4loz.  11  dwt.  bgr. 

24:  =  677  dwt.  +    5gr. 
20  =    28  oz.     +  17  dwt. 

--12=         21b         -f      4:0Z. 


70.  Question  third.  —  To  convert  a  given  compound  numher 
into  a  fraction  of  the  principal  unit. 

This  is  also  a  consequence  of  (68). 

Take,  for  example,  21b.  4:oz.  17  dwt.  5gr.  This,  reduced  to 
grains,  gives  13853  grs. ;  and,  by  the  tables  of  (66),  1  gr.  is  ^^^j 
of  ^^  of  -f\j  of  a  pound,  or  j^^^^  of  a  pound ;  the  required  frac- 
tion is  obviously  then  VWo^  ^^  ^  pound. 


COMPOUND   NUMBERS.  93 

Rule.  —  Commence  hy  reducing  the  given  compound  number 
into  units  of  the  lowest  denomination  which  it  contains;  then 
form  a  fractional  number  which  has  for  numerator  the  number 
thus  obtained,  and  for  denominator  the  ntimber  of  units  of  this 
lowest  denomination  which  the  principal  unit  contains. 

We  will  find,  by  this  method,  23  h.  55'  9"=  J^^  =^^ 
of  ail  hour. 

71.  Question  fourth. — Reciprocally j  given,  any  fractional 
number  of  the  principal  unit  of  a  certain  denomination,  to  con- 
vert it  into  a  compound  number. 

OPERATION. 

Given,  for  example,  |  of  a  mile  to  be  converted       5 
into  furlongs,  poles,  &c.     Since  each  mile  equals  eight       8 

furlongs,  f  of  a  mile  is  f  of  8  furlongs ;  equals  "^^  of  40  |  7 

1  furlong.     We  then  divide  40  by  7 ;  the  quotient,  5       5 

5,  expresses  obviously  the  furlongs,  and  the  remain-  40 

der,  5,  with  the   divisor,  7,   for  denominator,  is  a  200  |  7 
fraction  of  a  furlong  which  it  is  necessary  to  reduce       60     28 
to  poles.     Now,  1  furlong  equals  40  poles;  hence,  ^         4 

of  a  fur.  =  f  of  40  poles,  equals  — - —  of  1  pole.  Per-       — -  ^ 

7  Z2i  I  7 

forming  the  operations  here  indicated,  we  have  28  for        1    |  3 
quotient,  and  4  of  a  pole,  for  the  fraction  correspond- 
ing to  the  remainder ;   ^  o^  ^  P^^^  =  4  ^^  ^i  yards  =  ^  of  a 
yard,  which  is  equal  to  3^.   Hence,  the  required  compound  num- 
ber is  5  fur.  28  pis.  3|  yds. 

General  Rule.  —  In  order  to  convert  a  fractional  number 
of  any  principal  unit  into  a  compound  number,  obtain  first  the 
entire  number,  if  there  be  one,  contained  in  the  fraction;  you 
obtain  thus  a  certain  number  of  the  principal  units. 

Multiply,  then,  the  renfiainder  of  this  division,  by  the  number 
which  expresses  how  often  the  principal  unit  contains  the  next 
lower  subdivision,  and  divide  this  product  by  the  denominator 
of  the  given  number  ;  ice  thus  obtain  a  certain  number  of  units 


94  COMPOUND    NUMBERS. 

of  this  next  lower  suhdiviswn,  and  a  second  remainder.  Pro- 
ceed with  this  remainder  in  the  same  manner j  until  you  arrive 
at  a  result  with  no  remainder ^  or  exhaust  the  suhdivisions-of  the 
principal  unit. 

N,  B.  Principal  unit  can  apply  to  any  one  of  the  denomina- 
tions of  the  tables  (66),  which  has  itself  been  subdivided ;  that 
is  to  say,  every  subdivision  can  be  principal  unit  to  the  subdivi- 
sions below  it. 

72.  Remark  I.  —  The  operations  of  the  two  last  rules  can  serve 
as  verification  for  each  other.  Thus,  in  applying  (71),  to  the 
fractional  numbers  of  (70),  we  ought  to  reproduce  the  four  com- 
pound numbers  which  correspond.  In  the  same  manner,  we  can 
verify  the  result  of  (71),  by  means  of  the  rule  in  (70). 

73.  Remark  IL  —  The  principles  which  have  just  been  deve- 
loped would  be,  properly  speaking,  sufficient  to  permit  us  to  per- 
form the  four  fundamental  operations  of  arithmetic  upon  com- 
pound numbers. 

We  would  thus  pursue  the  following  method : 

1st.  Transform  the  compound  numbers,  each  one  into  a  frac- 
tion of  the  principal  unit  corresponding. 

2d.  Perform  upon  these  fractional  numbers  the  operation  pro- 
posed (^according  to  the  rules  of  the  calculus  of  fractions^  which 
icill  give  for  a  result  a  fractional  number. 

3d.  Convert  this  fractional  number  into  a  compound  number 
of  the  species  indicated  by  the  nature  of  the  question. 

Nevertheless,  since  the  direct  methods  of  performing  the  four 
fundamental  operations  upon  compound  numbers  give  rise  to  im- 
portant observations,  and  offer  for  the  theories  which  we  shall 
develop  later,  useful  applications,  we  will  proceed  to  discuss  them, 
as  simply  as  possible,  with  very  simple  examples. 

ADDITION   AND   SUBTRACTION. 

74.  —  1st.  Addition.  —  Place  (as  in  abstract  entire  numbers), 
the  given  numbers,  one  under  another,  so  that  the  units  of  the 
same  denomination  fall  under  each  other ;  after  whichy  make 


COMPOUND    NUMBERS.  95 

the  addition  of  the  units  contained  in  each  column ^  commencing 
on  the  right. 

If  the  sum  of  the  units  contained  in  a  column  exceeds  the 
number  which  expresses  how  many  times  the  unit  of  the  denomi- 
nation corresponding  is  contained  in  the  unit  of  the  next  higher 
denomination,  we  divide  the  sum  obtained  by  this  number  (69) ; 
we  obtain  thus  a  remainder,  (^possibly  0^)  which  we  write  below 
the  horizontal  line  drawn  under  all  the  columns,  and  a  quotient 
which  we  carry  to  the  units  of  the  following  column  ;  we  operate 
in  the  same  manner  upon  this  column,  and  upon  each  successive 
column.  (This  rule  is  obviously  established  by  the  same  reason- 
ing which  was  given  for  the  rule  in  simple  addition  of  abstract 
numbers). 

2d Subtraction.  —  Write  the   smaller   number  under  the 

greater,  so  that  the  units  of  the  same  denomination  fall  under 
each  other  ;  then  subtract  successively,  one  from  the  other,  the  units 
of  each  denomination,  commencing  with  the  loivest. 

When,  in  any  one  of  the  columns,  the  number  of  units  to  be 
subtracted  is  greater  than  the  number  from  which  it  is  to  be 
taken,  we  add  to  this  latter  (14),  a  unit  of  the  denomination  next 
higher,  converted  into  units  of  the  denomination  on  which  we  are 
operating ;  the  partial  subtraction  becomes  possible.  We  must 
take  care,  however,  to  augment  the  next  number  to  be  subtracted 
by  the  one  unit  borrowed  from  this  denomination. 

(This  rule  is  obviously  founded  on  the  reasoning  for  subtrac- 
tion of  simple  numbers). 

We  give  below  some  examples : 


£  s.     d. 

K>. 

oz. 

dwt. 

gr. 

17  13  4 

14 

10 

13 

20 

13  10  2 

13 

10 

18 

21 

10  17  3 

14 

10 

10 

10 

8   8  7 

1 

4 

4 

4 

3   3  4 

45 

•  0 

7 

7 

8  8 

13 

2 

12 

0 

54   1  4 

Proof 


23    32    0  Proof 


96  COMPOUND   NUMBERS. 


SUBTRACTION. 

£.     s.     d.                        mis.  fur.  pol.  yd.     ft.  in. 

67    13     8                       14      3      17  1      2  1 

49    17  11                       10      7      30  2  10 

7    15    9                         3      3      26  ^~l  3 


57    13     8  Proof  14      3      17      1      2      1  Proof 

The  methods  of  verification  are  the  same  as  in  abstract  num- 
bers, taking  care  to  preserve  the  relative  values  of  the  units 
carried  from  the  columns  of  higher  denominations  to  lower,  as  in 
the  verification  of  addition,  and  from  lower  to  higher,  as  in  sub- 
traction. 

MULTIPLICATION. 

75.  To  multiply  a  compound  number  by  a  simple  factor,  we 
consider  the  multiplication  of  each  denomination  of  the  compound 
number  as  a  separate  question ;  then  reduce  the  partial  products 
to  compound  numbers  by  (69),  and  add  these  compound  numbers 
by  last  article.  Or,  what  is  the  same  thing,  commence  on  the 
right  hand,  and  proceed  with  the  multiplication  as  in  simple 
numbers,  taking  care  to  preserve  the  proper  relative  values  be- 
tween the  successive  columns. 

Thus,  £4  13s.  M.  to  be  multiplied  by  9. 
£  .9.  d. 

4        13         3 

9 

41  19  3 
9  times  3  gives  27,  which  we  reduce  to  shillings,  giving  2  for 
quotient,  with  3  remainder;  set  down  the  3,  and  carry  the  2  to 
the  next  multiplication;  9  times  13  gives  119  ~  £5  19s.  We 
set  down  the  remainder,  and  add  the  5  to  the  product  of  9  by  £4, 
giving  £41.  If  we  have  one  denominate  number  to  be  multiplied 
by  another,  we  reduce  multiplicand  and  multiplier  to  fractional 
numbers  of  their  principal  units  by  (70);  then  multiply,  and 
reduce  the  result  back  to  the  compound  number  required  by  the 


COMPOUND    NUMBERS.  97 

question ;  or  we  may  simply  reduce  the  multiplier  to  such  a  frac- 
tion, and  proceed  as  in  the  first  example. 

Example.  —  £2  5s.  to  be  multiplied  by  101b  boz.  avoirdupois. 
We  may  either  multiply  |J  by  Le_5^  and  reduce  theiresult  to  pounds 
and  shillings,  or  we  may  multiply  £2  5s.  by  ^g^,  reducing  each 
result  separately. 

76.  Remarh.  —  It  xesults  obviously  from  this  mode  of  pro- 
ceeding, 

1st.  That  although  the  multiplier  is  a  denominate  number,  yet 
we  consider  the  principal  unit  of  this  factor  and  its  subdivisions 
as  abstract  numbers,  which  express  the  number  of  times  we  must 
take  the  multiplicand,  and  what  parts  of  it  we  must  take,  in  order 
to  obtain  the  required  result  \  but  we  preserve  always  in  the  mul- 
tiplicand its  essential  quality  of  concrete  number. 

2d.  That  all  the  partial  products  and  the  total  product  are 
always  of  the  same  nature  as  the  multiplicand. 

Certain  questions  of  Geometry,  however,  namely :  those  which 
have  for  their  object  the  measure  of  surfaces  and  volumes,  give 
rise  to  operations  which  form  exceptions  to  this  general  principle. 
The  considerations  on  which  these  are  founded  do  not  belong  to 
arithmetic. 

DIVISION. 

We  will  dwell  but  little  on  this  operation^  in  effecting  which, 
in  general,  it  is  better  to  apply  the  method  established  in  (73). 
Nevertheless,  we  will  consider  the  two  principal  cases  which  can 
present  themselves. 

77.  Ca&e  I.  —  In  which  the  dividend  and  divisor  are  com- 
pound numbers  of  the  same  species.  For  example  :  —  Required, 
How  many  yards  of  a  certain  work  can  we  have  executed  for 
£75  19s.  6d.,  if  one  yard  cost  £8  15s.  Qd.  ? 

It  is  clear  that,  for  the  resolution  of  this  question,  we  must 
determine  how  many  times  the  smaller  of  these  two  compound 
numbers  is  contained  in  the  greater.     This  is  effected,  1st,  by 


98  COMPOUND    NUMBERS. 

reducing  the  two  numbers  to  tlie  lowest  denomination  which 
enter  them ;  2d,  by  then  dividing  the  entire  numbers  thus  ob- 
tained one  by  the  other.  The  quotient  is  at  first  an  abstract 
number,  which,  according  to  the  enunciation  of  the  question,  can 
then  be  expressed  in  yards,  feet,  inches,  &c.  Converting  the  two 
given  numbers  to  pence,  we  find  18,233£7.  and  210Qd.  The  frac- 
tional number  then  will  be  ^^Wg ,  which  can  be  converted  into 
yards,  &c.,  by  rule  in  Art.  (71). 

78.  Case  II.  —  That  in  which  the  dividend  and  divisor 
are  of  different  sjjecies.  In  this  case,  whatever  be  the  question 
proposed,  the  quotient  must  express  principal  units  of  the  same 
species  as  the  dividend ;  since  it  is  necessary  that  the  dividend, 
considered  as  a  product,  must  be  of  the  same  species  as  one  of  its 
factors.  Eut  then,  the  compound  divisor,  being  converted  into  a 
fractional  number  of  the  principal  unit,  becomes  an  abstract 
number,  by  which  we  must  divide  the  dividend,  which  is  done  by 
multiplying  the  dividend  by  this  fraction  inverted  (60). 

79.  Remark  I.  —  We  conclude  from  the  above,  1st.  That  in 
every  division  of  compound  numbers,  if  the  two  numbers  are  of 
the  same  species,  the  quotient  is  considered  first  as  an  abstract 
number,  which  we  make  express  the  units  and  subdivisions  of 
units,  fixed  by  the  enunciation  of  the  question.  This  quotient  is 
to  be  the  multiplier  in  the  verification  of  the  operation  by  multi- 
plication. 

2d.  That  if,  on  the  contrary,  the  two  terms  of  the  division  are 
of  difi"erent  species,  the  quotient  expresses  necessarily  units  of 
the  same  species  as  the  dividend ;  while  the  divisor,  though  com- 
pound at  first,  is  to  be  regarded  as  an  abstract  number,  which 
plays  the  part  of  multiplier  in  the  verification  of  the  operation. 

80.  Remark  II.  —  So  far,  we  have  only  ^iven  one  method  of 
verifying  multiplication,  viz  :  the  method  by  division,  and  reci- 
procally. But  in  the  practice  of  the  operations  upon  compound 
numbers,  it  is  generally  more  convenient  to  verify,  1st,  Multipli- 
cation, by  doubling  one  of  the  two  factors,  and  taking  the  half 


COMPOUND    NUMBERS.  99 

of  tlie  other;  then  performing  the  operation  anew  with  the  re- 
sulting numbers.  2d,  Division,  by  doubling  the  two  terms  of 
the  division.  We  avoid  thus  the  difficulties  arising  from  the 
vulgar  fractions,  which  ordinarily  accompany  the  results  ar- 
rived at. 

It  is  evident  that  this  means  of  verification  can  also  be  em- 
ployed with  entire  abstract  numbers. 

EXERCISES. 

1.  Find  a  nifmber,  the  'i,  the  |,  |,  and  |  of  which,  added  to- 
gether, form  a  sum  which,  diminished  by  139,  gives  1289  for 
remainder. 

2.  A  reservoir  is  filled  by  four  difi'erent  pipes.  The  first  can 
fill  it  alone  in  5  hours ;  the  second  in  7  hours ;  the  third  in  9 
hours;  the  fourth  in  11  houi*s.  Required,  the  time  of  filling  the 
reservoir,  all  four  pipes  being  opened  at  onc6. 

3.  The  population  of  Asia  is  estimated  at  390,257,000  inhab- 
itants :  Required  the  population  of  Europe,  Africa  and  America; 
knowing  that  the  population  of  Europe  is  -^^  of  the  population 
of  Asia;  that  of  Africa,  -j^y  of  that  of  Europe;  and  that  of 
America,  j\  of  the  same. 

4.  The  sea  covers  i|  of  the  whole  surface  of  the  globe.  The 
surface  of  Asia  is  equal  to  ^^V  ^^  ^^^^  ^^  Europe ;  that  of  Africa 
is  ^^  of  the  same;  that  of  America,  y^^  ;  and  that  of  Oceanica, 
1^ ;  we  know,  besides,  that  Africa  has  a  superficies  of  13,450,000 
square  miles.  Calculate  the  superficies  of  the  other  parts  of  the 
world,  and  deduce  the  number  of  square  miles  in  the  whole 
surface. 

5.  Demonstrate  that,  by  adding  the  same  number  to  the  two 
terms  of  a  fractional  number,  we  obtain  a  result  which  approaches 
unity  more  as  the  number  added  is  greater.  Show  that  the  dif- 
ference between  the  result  and  unity  can  become  less  than  any 
given  quantity. 


100  DECIMAL   FRACTIONS. 

6.  Find  the  method  of  obtaining  the  greatest  common  divisor 
of  two  or  more  fractions.     Apply  to  the  fractions, 

3  5  7  19 

7.  Demonstrate  the  method  for  obtaining  the  least  common 
multiple  of  several  fractions. 


Apply  to  the  fractions,         -f^,  IJ,  !§. 

8.  What  is  the  greatest  common  multiple  of  the  fractions,  ||, 
-/p  and  If,  less  than  100,000. 

9.  What  will  be  the  price  of  a  piece  of  stuif,  23^'^  yards  long, 
each  yard  costing  £5  10s.  Qd.  ? 

10.  87  R).  10  0^.  5(7r.,  of  a  certain  material,  was  bought  for 
50£  lis.  del.     What  is  the  price  per  pound  ? 


CHAPTER  IV. 

» 

0/  Decimal  Fractions,  and  their  Principal  Properties —  Of  tlie 
Decimal  Systems  of  Compound  Numbers. 

I. -DECIMAL  FRACTIONS. 

81.  Introduction.  —  In  the  ordinary  system  of  numeration,  the 
most  simple  method,  and  the  most  convenient  one  of  subdividing 
unity,  is  the  suhdivision  into  successive  2Jarfs,  decreasing  in  a  ten- 
fold ratio.  From  this  mode  of  subdivision  result  fractions 
which  have  for  denominators  unit?/,  followed  hy  one  or  more 
7:eros^  and  these  fractions  we  call  decimal  fractions. 

This  mode  of  subdividing  unity  oflers  great  advantages,  inas- 
much as  it  reduces  immediately,  or  at  least  by  very  simple  trans- 
formations, all  the  operations  upon  fractional  numbers,  to  simple 
operations  upon  entire  numbers.  These  methods  we  will  develop 
f  fter  having  made  known  the  numeration  of  decimal  fractions; 


DECIMAL   FE ACTIONS.  101 

that  is,  their  nomenclature,  and  the  manner  of  writing  them  in 
figures. 

82.  Numeration  of  Decimals.  —  As,  by  increasing  unity  ten- 
fold, one  hundred-fold,  &c.,  successively,  we  form  new  units,  to 
which  we  give  the  name  of  tens,  hundreds,  thousands,  and  so- 
forth,  in  the  same  manner  we  conceive  unity  to  be  divided  into 
10  equal  parts,  which  we  call  tenths,  each  tenth  divided  into  10 
equal  parts,  which  we  call  hundredths,  (because  the  principal 
unit  contains  10  times  10,  or  100  of  these  new  parts  or  units) ; 
then  each  hundredth  divided  into  10  equal  parts,  called  thou- 
sandths, and  so  on;  thus  giving  ten  thousandths,  hundred  thou- 
sandths, &c. 

In  the  second  place,  it  results,  (5),  from  the  fundamental 
principle  of  the  written  numeration  of  entire  numbers,  that  the 
figures,  proceeding  from  right  to  left,  have  their  relative  value 
increased  tenfold  for  each  place  to  the  left,  and  decreased  ten- 
fold, going  from  left  to  right.  Whence  it  follows,  that  if  to  the 
right  of  an  entire  number  written  in  figures  we  place  new  figures, 
taking  care  always  to  distinguish  them  by  any  sign  whatever,  a 
comma  or  point  for  example,  from  the  entire  number,  we  shall 
thus  represent  successive  parts  of  unity,  decreasing  tenfold  to 
the  right ;  that  is,  tenths,  hundredths,  thousandths,  &c. 

Thus,  the  collection  of  figures,  24,75,  expresses  24  luiits,  7 
tenths,  and  5  hundredths;  5,478  equals  5  units,  4  tenths,  7  hun- 
dredths, and  8  thousandths. 

83.  Let  it  be  required  to  enunciate  in  ordinary  language  the 
number  56,3506.  This  number  can  at  first  be  enunciated  56 
units^  3  tenths,  5  hundredths,  0  thousandths,  and  6  ten  thousandths. 
But  3  tenths  are  equal  to  30  hundredths,  or  300  thousandths,  or 
3000  ten  thousandths;  in  the  same  manner,  5  hundredths  are 
equal  to  50  thousandths,  or  500  ten  thousandths  The  number 
can  then  be  enunciated  56  units,  and  3506  ten  thousandths. 

Thus,  in  order  to  enunciate  in  ordinary  language  a  decimal 
fractional  number  written  in   figures,  we  must  enunciate  sepa^ 
ratelij  the  entire  part,  and  then  enunciate  the  part  ichich  is  to 
0  * 


102  DECIMAL    FRACTIONS. 

tlie  right  of  tlie  comma,  as  an  entire  rnnnhcr,  aiving  at  the  close 
the  name  of  the  unit  of  the  last  decimal  aubdivisicrii. 

Thus,  7,49305  represents  7  uniu  and  49305  hundred  thou- 
sandths. In  the  same  manner,  249,007,056  represents  249  units 
and  7056  millionths.  We  can  also,  if  we  wish,  include  in  one 
single  enunciation  the  entire  as  well  as  the  decimal  part 

Take,  for  example,  the  number  56,3506.  As  one  unit  equals 
10  tenths,  or  100  hundredths,  1000  thousandths,  &c.,  it  follows, 
that  56  units  are  equal  to  560000  ten  thousandths ;  and,  conse- 
quently, 56,3506  represents  563506  ten  thousandths.  That  is, 
we  must,  after  enunciating  the  number  as  if  it  had  no  comma, 
place  at  the  end  of  the  number  thus  enunciated,  the  name  of 
the  last  subdivision.  It  is  customary,  however,  to  enunciate  the 
entire  part  separately. 

We  will  indicate  a  method  for  enunciating  the  decimal  part, 
which,  in  general,  is  more  convenient  in  practice.  After  an- 
nouncing the  entire  part,  as  we  have  just  said,  separate  mentally 
the  decimal  part  into  periods  of  three  figures,  beginning  at  the 
comma,  (the  last  period  having  often  only  one  or  two  figures) ; 
enunciate  then  each  period  or  division  separately,  and  place  at 
the  end  of  each  partial  enunciation  the  name  of  the  last  unit 
of  the  period. 

Example. — The  number,  2,74986329,  is  enunciated;  2  units, 
749  thousandths,  863  millionths,  29  hundred  millionths. 

84.  Reciprocally,  we  propose  to  write  in  figures  a  decimal 
fraction  enunciated  in  ordinary  language. 

Required  to  write  in  figures  the  number;  twenty-nine  units, 
three  hundred  and  fifty  four  thousandths.  Write  first  the  entire 
part,  29 ;  then,  as  300  thousandths  are  equal  to  3  tenths,  and 
50  thousandths  equal  5  hundredths,  place  a  comma  to  the  right 
of  29,  and  write  successively  the  numbers  3,  5,  and  4 ;  we  thus 
have  29,354. 

Iq  like  manner,  one  hundred  and'  nine  units,  two  thousand 
and  three  ten  thousandths,  are  written  109,2003. 


DECIMAL   FRACTIONS.  103 

Required,  again,  to  write  the  number  eight  units,  thirty-seven 
thousandths.  As  thirty  thousandths  make  3  hundredths,  and  as 
there  are  no  tenths  in  the  number  enunciated,  we  write  8,087; 
that  is  to  say,  we  make  the  same  use  of  the  0  in  both  these  last 
cases  as  in  whole  numbers,  placing  it  here  to  the  right  of  the 
comma,  to  take  the  place  of  the  tens  which  are  wanting,  and  to 
give  the  figures  which  follow  their  true  value. 

GrENERAL  KuLE.  —  In  order  to  write,  in  figures,  a  decimal 
enunciated  in  ordinary  language,  commence  hy  writing  the  entire 
part,  and  after  it  a  comma  or  point;  then  write  successively,  to 
the  right  of  this  point,  the  figures  luhich  represent  the  tenths,  hun- 
dredths, &c.,  included  in  the  number,  taking  care  to  replace  hy 
zeros  the  different  orders  of  unit^s  which  are  wanting.  If  there 
is  no  entire  part,  lorite  a  0  to  take  the  place  of  it,  and  proceed 
v:ith  the  decimal  part  as  before. 

Thus,  seventeen  hundredths  are  represented  by  0,17;  one 
hundred  and  twenty-five  ten  thousandths  by  0,0125. 

It  may  happen  that,  in  the  enunciation  of  the  number,  the 
entire  part  is  not  distinguished  from  the  decimal  part.  We  must 
then  write  the  number  as  if  it  expressed  entire  units,  and  then 
place  a  point  so  that  the  last  figure  to  the  right  shall  express  the 
units  of  the  last  subdivision  of  the  number  enunciated. 

For  example,  in  order  to  write  the  number  four  thousand.  Wo 
hundred  and  fourteen  hundredths,  write  first  4214 ;  and,  as  the 
last  figure  must  express  hundredths,  place  the  comma  between 
the  2  and  1,  giving  42,14.  Two  hundred  and  fifty-three  thou- 
sand and  twenty-nine  ten  thousandths,  are  represented  by 
25,3029. 

85.  Decimal  fractions  placed  under  the  form  of  vulgar  frac- 
tions. A  fraction  being  composed  of  two  terms,  the  numerator 
and  the  denominator,  the  comma  serves,  in  the  method  which  we 
have  just  developed,  to  indicate  the  denominator,  which  is  equal 
to  unity,  followed  by  as  many  zeros  as  there  are  decimal  figures ; 
that  is,  figures  to  lAe  right  of  the  comma.     The  numerator,  we 


104  DECIMAL   FRACTIONS. 

have  seen,  is  composed  of  the  collection  of  figures  to  the  right 
of  the  comma.  Or,  if  we  consider  the  entire  part" as  reduced  to 
a  fraction,  the  numerator  is  then  the  number  given,  with  the 
comma  stricken  out.  Thus,  the  number,  23,6037,  put  under  the 
form  of  a  vulgar  fraction,  is  23fgO_3j7_^  qj.^  2_35__o^3J7^  The  number, 
2,00409,  is  equal  to  2^/oVoo.  or,  f  gg-J-oa.  Finally,  0,0002154, 
is  equal  to  _^§f54^^.  Reciprocally,  2y§-3_o,  or,  f-Q§-3,  is  equal 
to  2,053 ;    VVoVcf  is  equal  to  17,2049. 

These  two  transformations  are  of  continual  use  in  the  calculus 
of  decimal  fractions. 

86.  CJianging  the  place  of  the  point.  —  If,  in  a  decimal  frac- 
tion, we  advance  the  point  one  or  more  places  to  the  right,  we 
multiply  the  number  by  10,  100,  1000,  &c. ;  and  if,  on  the  con- 
trary, we  place  it  one  or  more  places  farther  to  the  left,  we  divide 
the  number  by  10,  100,  1000,  &c. 

For,  let  the  number  be  153-07295. 

Suppose  we  advance  the  point  three  places  to  the  right,  which 
gives  153072-95.  The  two  numbers  are  now  ^f^J2§§^,  and 
*^  Wo^^^-  ^ow,  the  denominator  of  the  second  number  is  1000 
times  smaller  than  that  of  the  first,  while  the  numerator  is  the 
same.  Then,  the  second  fraction  is  1000  times  greater  than  the 
first.  On  the  contrary,  remove  the  point  two  places  towards  the 
left,  it  becomes  1-5307295,  or,  1550129.5^  a  fraction  evidently 
100  times  smaller  than  the  given  one.  We  could  establish  the 
same  thing  by  reasoning  thus  :  —  By  changing  the  place  of  the 
point,  the  value  of  each  figure  becomes  10,  100,  1000,  &c., 
times  greater  or  smaller.  Thus,  in  comparing  153072-95,  with 
153-07295,  we  see  that  the  figure  3,  which  expresses  in  the 
latter  simple  units,  expresses  now  thousands;  the  figure  5,  to  the 
left  of  the  figure  3,  which  expressed  tens,  represents  now  tens  of 
thousands  ;  and  the  same  with  the  other  figures. 

87.  Zeros  placed  to  the  right  of  a  decimal  fraction. 

By  annexing  any  number  lahatevcr  of  zeros  to  the  right  of  a 
decimal  fraction,  v-e  do  not  change  its  value. 


DECIMAL   FRACTIONS  105 

Thus,  3-415  is  equivalent  to  3-4150,  3-41500  .  .  .  . ;  for  these 
numbers  can  be  (85),  put  under  the  form, 

3415       3  415  0       3  4_15  00 

1000'  Toooo>    100000?  •  •  •  •; 

Now,  the  last  two  fractions  are  nothing  more  than  the  first, 
with  its  two  terms,  multiplied  by  10,  100,  which  (43),  does  not 
change  its  value.     Then,  &c 

Or,  we  may  observe  that  zeros,  placed  to  the  right  of  decimal 
figures  already  written,  do  not  change  their  value  ]  and,  as  these 
zeros  have  no  value  of  themselves,  the  fraction  remains  always 
the  same.  As  the  value  of  a  figure  in  a  decimal  fraction  depends 
entirely  on  the  number  of  places  it  is  distant  from  the  point,  it 
is  obvious  that  we  do  alter  this  value  by  prefixing  zeros  between 
the  decimal  point  and  the  first  decimal  figure. 

88.  Reduction  of  several  decimal  fractions  to  the  same  deno- 
minator. 

The  principle  which  has  just  been  established,  gives  us  a  me- 
thod of  reducing  several  decimal  fractions  to  the  same  number 
of  decimal  fgures,  without  changing  their  value ;  or,  in  other 
terms,  to  the  same  denominator. 

For  example,  the  fractions 

12-407    I  0-25      I  7-0456  |  23-4 
are  equal  to        12-4070  |  0-2500  |  7-0456  |  23-4000. 

They  have  10000  for  common  denominator.  These  prelimi- 
nary ideas  being  established,  we  pass  to  the  four  fundamental 
operations  upon  decimal  fractions. 

Addition  and  Subtraction. 

89.  We  perform  the  addition  of  decimal  fractions  in  the  same 
manner  as  ice  do  that  of  entire  vumhers,  offer  reducing  them  all 
to  the  same  denominator,  and  we  pfoint  off  in  the  result  as  man?/ 
decimal  places  as  there  are  in  any  one  of  the  reduced  numbers, 
or  the  greatest  number  which  any  one  of  the  given  numbers  con- 
tains. 


lOG  DECIMAL   FRACTIONS. 

A  single  example  will  suffice  to  illustrate  and  make  plain  this 
rule. 

Given,  to  add  the  numbers 

32-4056  I  245-379  |  12-0476  |  9-38  |  and  459-2375. 

32-4056 
245-3790 

12-0476 

9-3800 

459-2375 


758-4497 


121-2210    Verification. 

We  write,  first,  one  zero  to  the  right  of  the  second  number, 
and  two  to  the  right  of  the  fourth ;  we  then  place  the  numbers 
thus  prepared,  one  under  another,  so  that  the  units  of  the  same 
order  correspond,  and  then  make  the  addition  in  the  ordinary 
manner.  We  find  for  result,  7584497;  or,  separating  the  four 
figures  to  the  right,  758-4497;  because  the  numbers  added  ex- 
press units  of  the  order  of  ten  thousandths. 

In  practice,  we  can  dispense  with  writing  the  zeros  to  the  right 
of  the  numbers,  which  contain  fewer  decimal  places  than  the 
others,  provided  we  take  care  to  arrange  the  units  of  the  same 
order  in  the  same  column. 

Subtra-ction  is  'performed  in  the  same  manner  as  in  entire 
numbers,  after  ice  have  reduced  the  decimals  to  the  same  deno- 
minator (88). 

Example.  — G^'wi^w,  to  subtract  23-0784  from  62-09. 

62  0900 
23-0784 


39-0116 


62-0900    Verification. 

We  write  two  zeros  to  the  right  of  the  62-09,  which  gives 
62-0900;   we  then  perform  subtraction   in  the  usual  manner, 


DEC13IAL   FRACTIONS.  107 

taking  care  to  separate  four  decimal  figures  to  the  right  of  the 
result. 

These  methods  are  obviousl3'  founded  upon  the  fact  that  the 
units  of  different  orders,  in  decimal  fractions,  having  the  same 
relations  of  magnitude,  one  to  the  other,  as  in  entire  numbers, 
we  have  the  same  operations  to  be  performed  with  the  figures  to 
be  carried  as  in  entire  numbers. 

Multiplication. 

90.  In  order  to  perform  this  operation,  multipli/  the  two  given 
numbers  one  hi/  the  other,  without  regarding  the  comma  or  point 
ivhich  they  contain;  then  separate  hy  a  point,  from  the  right  of 
the  product  thus  ohtained,  as  many  decimal  figures  as  there  are 
in  both  factors. 

Required,  for  example,  to  multiply  85-407  by  12-54.  We  find 
first  for  the  product  of  the  two  numbers,  the  points  being  disre- 
garded, 44400378.  Pointing  ofi",  then,  on  the  right  of  the  pro- 
duct, 3  -f  2,  or  5  figures,  we  obtain  for  the  required  product, 
444-00378.  In  order  to  see  the  reason  of  this  method,  we  re- 
mark, that  the  two  given  numbers  are  equal  to  (160),  jVqV^  ^^^ 
Wo"*-     Whence  we  deduce  the  product  by  the  rule  in  (57), 

35407x1254    ,,   ,  .   ,  .,  .  ,  ,  .  ,     , 

-TKc^ — TKcT )  t"^*  IS  to  say,  it  IS  necessary,  1st,  to  multiply  the 

two  numbers,  disregarding  the  point ;  2d,  to  divide  this  product 
by  100000,  or  unity,  followed  by  as  many  zeros  as  there  are  de- 
cimal figures  in  the  two  factors,  which  is  equivalent  to  separating 
5  decimal  figures  on  the  right  of  the  product.  The  method  is 
thus  justified.  Or,  we  may  reason  thus  :  by  removing  the  point 
from  the  multiplicand,  we  multiply  it  by  1000 ;  since,  at  first,  it 
expresses  thousandths,  but  after  the  multiplication,  principally 
units;  then,  the  product  is  1000  times  too  great.  In  the  same 
manner,  by  removing  the  point  from  the  multiplier,  we  render  it 
100  times  greater.  Thus,  by  the  suppression  of  both  points,  the 
product  is  rendered  100000  times  too  great ;  then,  in  order  to 
bring  it  back  to  its  just  value,  it  must  be  divided  by  100000,  or 


108  DECIMAL   FRACTIONS. 

five  figures  must  be  pointed  off  for  decimals  on  the  right.  The 
reasoning  would  obviously  be  the  same,  whateve/be  the  number 
of  decimals  in  the  two  factors.  It  can  happen  that  one  of  the 
two  numbers,  only  contains  decimals.  In  this  case,  we  point  off, 
on  the  right  of  the  product,  as  many  decimal  figures  as  there  are 
in  this  number.  The  demonstration  is  too  easy  and  obvious  to 
detain  us. 

We  will  find,  according  to  these  rules, 

1st.  The  product  of  4-057  by  9-503,  is  38-553671. 

2d.  The  product  of  4-0015  by  29,  is  116-0435. 

3d.  The  product  of  0-03054  by  0-023,  is  0-00070242. 

N.  B.  This  last  example  deserves  some  attention.  Suppress- 
ing the  point  in  the  two  factors,  and  performing  the  multiplica- 
tion, we  find  for  a  product,  70242  ;  but,  as  there  are  five  decimals 
in  the  multiplicand,  and  three  in  the  multiplier,  there  must  be 
eight  of  them  in  a  product  which  contains  only  five  figures.  In 
order  to  remove  the  difficulty,  we  observe,  that  as  the  product 
ought  to  express  units  of  the  8th  order  of  decimals,  it  suffices  to 
write,  on  the  left  of  70242,  zeros  in  such  number  that,  the  point 
being  placed  on  the  left  of  them,  the  last  figure  to  the  right  shall 
occupy  the  8th  decimal  rank.  We  write  three  zeros  then  on  the 
left,  besides  one  for  the  entire  number,  and  obtain  0-00070242. 

Division. 

91.  Two  principal  cases  present  themselves.  Either  the  divi- 
dend and  divisor  have  the  same  number  of  decimals,  or  this 
number  is  different.  In  the  first  case,  suppress  the  point  in  the 
dividend  and  in  the  divisor  ;  then  operate  upon  the  entire  num- 
bers which  result  from  it.  according  to  the  ordinary  ride  of 
division. 

In  the  second,  commence  by  reducing  the  two  given  numbers 
to  the  same  number  of  decimal  places,  or  to  the  same  denomina- 
tor.    The  second  case  thus  becomes  the  first. 

First  Case.— Required  to  divide  47-359  by  8-234.  These  two 
numbers  can  be  put  under  the  forms  (85),  VoVu?  IMJ-  I^ividing 


DECIMAL   FRACTIONS.  109 

them  one  by  the  other,  according  to  rule  for  the  division  of  frac- 
47359      1000      47359x1000      47359 
tions  (59),  we  have  -looo"  ^  8 234  =    8234x1000  =  ^234' 
suppressing  the  factor,  1000,  common  to  the  two  terms. 

We  see,  then,  that  the  quotient  required  is  equal  to  that  of 
the  two  given  numbers  with  the  point  removed ;  and  the  rule 
above  is  proved.  We  can  also  say,  the  two  decimal  fractions 
having  the  same  denominator,  if  we  suppress  the  point,  we  mul- 
tiply the  two  terms  of  the  division  by  the  same  number,  1000; 
then,  the  value  of  the  quotient  remains  the  same.  The  division 
of  47359  by  8234,  gives  for  the  entire  part  of  the  quotient,  5, 
and  for  remainder,  6189;  thus  the  total  quotient  is,  5|^||. 

92.  Valuation  of  the  quotient  in  decimals. — The  vulgar  frac- 
tion, which  accompanies  the  entire  part  of  the  quotient,  having 
terms  pretty  large,  it  is  difficult  to  value  it  in  its  present  state ; 
moreover,  it  is  natural  to  endeavour  to  express  it  in  parts  of  the 
same  species  as  the  given  numbers.  We  arrive  at  this  now  by 
the  rule  in  (63) : 

4735918234 


61890   5-7516395 


42520 


13500 
~52660 

~^5'60 

"78580 

~uuo 
"3570" 

After  obtaining  the  entire  part,  5,  of  the  quotient,  in  order  to 
make  the  remainder,  6189,  express  tenths,  we  multiply  it  (63), 
by  10 ;  this  we  effect  by  placing  a  0  on  its  right ;  then  we  divide 
61890  by  8234;  the  quotient,  7,  expresses  then  tenths;  and  we 


110  DECIMAL   FRACTIONS. 

write  it  to  the  right  of  the  figure  5,  with  a  point  before  it.  To 
the  right  of  the  new  remainder,  4252,  we  place  a  0,  in  order  to 
convert  it  into  hundredths ;  we  then  divide  42520  by  8,  which 
gives  the  quotient,  5 ;  this  we  place  on  the  right  of  7,  and  annex 
another  0  to  the  remainder,  1350 ;  and  so  on,  until  we  have  ob- 
tained the  number  of  decimal  places  which  the  enunciation  ques- 
tion giving  rise  to  the  decision  demands. 

GrENERAL  KuLE.  —  In  order  to  express,  in  decimals,  the  quo- 
tient of  the  division  of  two  decimal  numbers  of  the  same  denom 
minator,  or  (which  is  the  same  thing  after  the  suppression  of  the 
point),  of  any  two  entire  numhers  ivhatever, 

Commence  hy  determining  the  entire  part  of  the  quotient ^ 
(which  can  be  0),  and  lorite  a  point  after  it. 

Annex  a  zero  to  the  right  of  the  remainder  ;  divide  the  num- 
ber thus  formed  by  the  divisor ;  then  place  the  quotient  on  the 
right  of  the  point.  Annex  aO  to  the  right  of  the  new  remainder, 
and  perform  the  division  by  the  same  divisor  ;  write  the  quotient 
on  the  right  of  the  two  first.  Continue  thus  until  you  have  the 
number  of  decimals  requi7-ed. 

93.  Remark  on  these  approximations.  —  In  the  preceding  ex- 
ample, we  have  carried  the  operation  as  far  as  the  seventh  deci- 
mal figure,  in  order  to  establish  some  principles  upon  the  different 
degrees  of  approximation  which  can  be  obtained  by  the  develop- 
ment of  a  number  into  decimals. 

By  taking  at  first  only  the  two  first  decimal  figures,  we  have 
5-75  for  the  value  of  the  quotient,  to  within  less  than  0*01,  since 
the  part  neglected  is  obviously  less  than  the  unit  of  this  order 
of  decimals.  Again,  as  this  neglected  part  is  less  than  0-002, 
oy—XqjOT  -gj^,  it  follows,  that  5-75  expresses  the  value  of  the 
quotient  to  within  less  than  ^i^. 

Now,  if  we  take  the  three  first  decimal  figures,  we  have  5*751 
for  the  value  of  the  quotient,  to  ivithin  less  than  0*001,  since  the 
part  which  we  neglect,  0*00063  ....  is  less  than  0*001.  But 
here  we  must  make  an  important  observation.     As  the  figure  6 


DECIMAL   FRACTIONS.  Ill 

exceeds  5,  it  follows  that  0-0006  exceeds  0-0005,  or  a  Jialfunit 
of  the  order  tJiousandths  ;  then,  by  taking  5-752,  instead  of  5-751, 
for  the  value  of  the  quotient,  we  commit  an  error  in  thus  taking 
more  than  the  true  value,  less  than  is  committed  when  we  take 
5-751  for  this  value;  and  we  can  say  that  5*752  expresses  the 
quotient,  not  only  to  within  less  than  0-001,  but  to  within  less 
than  the  half  of  O'OOl. 

Generally,  whenever  the  figure  which  follows  that  one  at  which 
we  wish  to  stop  in  the  divisioUj  is  less  than  5,  tee  preserve  the 
figure  obfainedj  and  we  then  have  the  value  of  the  quotient  to 
within  less  than  a  half  unit  of  the  denomination  at  which  we 
stop.  If  on  the  contrary,  the  figure  which  follows  is  equal  or 
greater  than  5,  it  is  best  to  increase  hy  one  unit  the  last  figure 
obtained,  in  order  to  obtain  a  value  nearer  the  quotient;  the 
error  committed  is  an  error  of  excess,  but  it  is  less  than  a  half 
unit  of  the  order  at  which  we  stop  the  operation. 

Thus,  in  the  example  above,  we  have  successively  for  the  quo- 
tient of  the  proposed  division,  5-752,  too  great  by  less  than  a 
half  thousandth  ;  5-7516,  too  small  by  less  than  a  half  ten  thou- 
sandth; 5-75164,  too  great  by  less  than  a  half  hundred  thou- 
sandth; 5,751640,  too  great  by  less  than  a  half  millionth.  "We 
will  add,  that  when  we  have  arrived  at  any  decimal  figure  what- 
ever, in  the  operation  performed,  the  last  remainder  obtained 
shows  whether  the  following  figure  of  the  quotient  is  greater  or 
less  than  5,  without  necessarily  calculating  this  figure. 

If  the  remainder  is  less  than  half  the  divisor,  the  following 
figure  of  the  quotient  will  necessarily  be  less  than  5. 

If  this  remainder  is  equal  to,  or  greater  than  the  half  of  the 
divisor,  the  next  figure  of  the  quotient  will  be  equal  to,  or  greater 
than  5. 

Thus,  in  the  example  which  we  have  just  discussed,  the  eighth 
figure  of  the  quotient  must  be  less  than  5 ;  for,  the  remainder  at 
which  we  stopped,  3570,  is  obviously  less  than  the  half  of  the 
divisor,  8234. 


112  DECIMAL   FRACTIONS. 

We  have  here  given  the  whole  theory  of  approximations  in 
the  valuation  of  fractional  numbers  in  decimals. 

94.    Case  Second.  —  This  divides  itself  into  two  others  : 

Firstly,  —  The  dividend  contains  fewer  decimal  figures  than 
the  divisor.  We  write  on  the  right  of  the  dividend  the  number 
of  zeros  necessary  to  reduce  the  two  terms  of  the  division  to  the 
same  number  of  decimal  places ;  and  the  question  is  solved  by 
Case  First  without  farther  modification. 

For  example  : — Required,  to  divide  2-405  by  0-03497.  Placing 
two  zeros  to  the  right  of  the  dividend,  which  gives  2-40500; 
then,  suppressing  the  comma  in  both  numbers,  we  perform  the 
division  of  the  two  resulting  numbers,  240500,  and  3497,  ac- 
cording to  the  rules  in  (91  and  92).  We  find  thus  the  value  of 
the  quotient  to  within  less  than  -0001,  to  be  68-7732. 

Secondly,  —  IVie  dividend  has  more  decimal  figures  than  the 
divisor  ;  ive  can  then  employ  two  methods. 

1st.  Required  to  divide  3*470456  by  1-027.  If  we  suppress 
the  point  in  the  divisor,  thus  rendering  it  1000  times  as  great, 
and  if  we  advance  the  point  in  the  dividend  three  places  to  the 
right,  rendering  it  thus  also  1000  times  as  great  as  at  first,  the 
quotient  of  the  division  of  these  two  numbers  resulting,  will  be 
the  same  as  that  of  the  given  numbers.  The  question  is  thus 
reduced  to  dividing  3470456  by  1027. 

3470-45611027 


3894      13-379217 


8135 
"9466 
~2230 


1760 
T3'30 

"Hi 


DECIMAL   FRACTIONS.  113 

After  finding  the  entire  part,  3,  of  tlie  quotient,  and  the  re- 
mainder, 389,  instead  of  placing,  as  in  (92),  a  zero  to  the  right 
of  this  remainder,  we  bring  down  the  figure  4,  which  expresses 
tenths,  and  perform  the  division,  obtaining  for  quotient,  3,  which 
we  place  on  the  right  of  the  first,  separating  .them  by  a  point; 
we  then  bring  down  to  the  remainder,  813,  the  figure  5,  which 
expresses  hundredths;  and  we  continue  thus,  until  we  have 
brought  down  all  the  decimal  figures  which  are  contained  in  the 
dividend.  When  we  reach  the  remainder,  223,  we  place  a  zero 
on  the  right  of  it,  and  operate  as  in  case  first.  We  see  that  this 
method  consists  in  suppressing  the  point  in  the  divisor,  taking 
care  to  remove  it  in  the  dividend  as  many  places  to  the  right  as 
there  are  decimals  in  the  divisor;  then,  in  operating  upon  the 
resulting  numbers,  as  in  the  first  case,  with  this  difi"erence,  that 
instead  of  annexing  at  first  zeros  to  the  right  of  the  diff"erent  re- 
mainders, we  commence  by  bringing  down  successively  all  the 
decimal  figures  of  the  dividend. 

2d.  We  take  the  same  example,  and  commence  by  writing  to 
the  right  of  the  divisor  three  zeros ;  that  is  to  say,  the  number 
of  zeros  necessary  to  reduce  the  two  terms  to  the  same  number 
of  decimal  places. 

We  have  then  to  divide  3470456  by  1027000. 

347056  1 1027(000 

389456 1 3-379217 

~81356 


9466 


2230 
"1760 


7330 
lil 


In  order  to  determine  the  entire  part  of  the  quotient,  we  com- 
mence by  applying  the.  rule  of  (38),  for  the  division  of  entire 


114  DECIMAL   FRACTIONS. 

numbers,  when  the  divisor  is  terminated  by  zeros,  We  obtain 
thus  the  quotient,  3,  and  the  remainder,  389456.  Now,  in  order 
to  find  the  tenths  figure,  we  remark,  that  instead  of  multiplying 
the  remainder  by  10,  (i.  e.)  placing  a  0  to  ttie  right  of  it,  wc  can 
divide  the  divisor  by  10 ',  that  is,  suppress  one  0  on  its  right. 
Performing  then  the  division,  we  have  3  for  quotient,  expressing 
tenths,  and  the  remainder,  81356.  In  the  same  manner,  instead 
of  putting  a  0  to  the  right  of  this  remainder,  we  suppress  a  se- 
cond 0  on  the  right  of  the  divisor,  and  divide  81356  by  10270 ; 
applying  still,  if  we  wish,  the  rule  of  (38).  We  obtain  thus  the 
new  quotient,  7,  and  the  remainder,  9466. 

Suppressing  the  last  0  on  the  right  of  the  divisor,  we  divide 
9466  by  1027 )  this  gives  the  quotient,  9,  and  the  remainder,  223. 
Setting  out  from  this  remainder,  we  follow  the  rule  in  (92),  in 
order'  to  obtain  the  remaining  decimal  figures.  This  second  me- 
thod is  obviously  less  simple  than  the  first;  and  we  mention  it, 
because  it  gives  us  the  opportunity  of  showing  how  to  operate 
when  we  have  zeros  to  annex  to  the  remainders  of  a  division,  of 
which  the  divisor  is  terminated  by  one  or  more  zeros. 

95.  Particular  Cases.  —  When  there  are  no  decimal  places  in 
one  of  the  terms  of  the  division  —  For  example,  we  can  have 
51-47876  to  be  divided  by  849,  or  3145  to  be  divided  by  23-479. 

In  the  first  of  these  examples,  we  would  proceed  according  to 
the  first  method  indicated  in  (94),  under  the  head  secondly. 

In  the  second,  we  suppress  the  point  in  the  divisor,  and  annex 
to  the  dividend  as  many  zeros  as  there  are  decimals  in  the  divi- 
sor. This  is  the  same  thing  as  multiplying  both  terms  by  the 
same  number.  These  cases  are  too  simple  to  demand  farther  de- 
velopment. 

Conversion  op  Vulgar  Fractions  into  Decimals. 

96.  We  have  seen  in  (92)  how  we  are  led  to  convert  a  vulgar 
fraction  into  a  decimal.  This  operation  forms  an  essential  part 
of  the  theory  of  the  division  of  decimal  fractions.  But  we  will 
make  here  an  important  observation,  which  shall  serve  us  in  the 


DECIMAL  FRACTIONS.  115 

exposition  of  tlie  properties  of  decimal  fractions,  whicli  we  have 
to  establish  hereafter. 

This  observation  consists  in  this,  that  instead  of  placing  zeros 
to  the  right  of  the  different  remainders  which  we  obtain  hy  apply- 
ing the  rule  of  (92),  we  can  place  at  once  these  zeros  on  the  right 
of  the  dividend,  and  perform  the  division  of  the  resulting  num- 
ber by  the  divisor,  taking  care  to  place  the  point  in  the  place  to 
which  it  belongs  in  the  quotient. 

In  order  to  establish  this  second  method  of  proceeding,  we 
take  the  example,  \^,  and  write  out  both  methods. 


130 

47 

13000000  47 

360 

0-276595 

360  0-276595 

310 

310 

280 

280 

450 

450 

270 

270 

35 

35 

In  the  first  method,  after  writing  a  zero  in  the  quotient,  to  take 
the  place  of  the  entire  number,  we  annex  a  zero  to  the  numera- 
tor, 13,  of  the  fraction,  in  order  to  obtain  the  tenths;  we  then 
place  another  zero  to  the  right  of  the  remainder  of  this  division, 
in  order  to  obtain  hundredths,  and  so  on,  until  the  total  number 
of  zesos  thus  successively  brought  down  is  six.  In  the  second 
method,  we  multiply  the  numerator  13  by  1000-000  first,  and 
then  perform  the  division.  It  is  obvious  that  the  quotient  thus 
obtained  difi'ers  from  that  obtained  by  the  first  method  of  pro- 
ceeding, in  being  1000000  times  greater,  and  that  we  reduce  it 
to  its  true  value  by  dividing  it  by  1000000,  or  by  pointing  off 
six  decimal  figures  on  the  right. 


116  DECIMAL   SYSTEM   OF   WEIGHTS,    &C. 

DECIMAL  SYSTEM  OF  WEIGHTS,  MEASimES,  AND 
COINS. 

Having  now  discussed  the  four  fundamental  operations  of 
arithmetic  in  their  application  to  decimal  fractions,  we  can  ap- 
preciate the  advantages  which  the  calculus  of  decimal  fractions 
presents  over  that  of  vulgar  fractions,  and  are  prepared  to  judge 
how  important  it  is  to  establish  a  decimal  system  of  weights, 
coins,  and  measures.  In  the  United  States  we  have  the  decimal 
system  of  coins  in  the  Federal  money.  In  France,  the  decimal 
system  of  weights,  coins,  and  measures,  has,  after  many  efforts, 
been  established,  in  spite  of  the  obstacles  occasioned  by  ignorance 
and  prejudice.  We  give  these  decimal  systems,  with  a  few  ex- 
amples, in  order  to  illustrate  their  advantages  over  the  ordinary 
systems,  with  their  irregular  subdivisions. 

97.  The  denominations  of  the  currency  of  the  United  States 
are  Eagles,  Dollars,  Dimes,  Cents,  and  Mills,  (the  last  three 
terms  expressing  their  relative  values  to  the  dollar  by  their  deri- 
vation). 

TABLE. 

The  Eagle  is  divided  into 10  dollars. 

The  Dollar  10  dimes. 

The  Dime 10  cents. 

The  Cent 10  mills. 

The  dollar  sign  being  ($),  we  would,  for  example,  write  56 
dollars,  57  cents,  and  5  mills,  simply  $56-575.  In  order  to  make 
the  comparison,  if  we  wished  to  write  £15  10s.  6f7.  in  parts  of  a 
pound,  we  would  have  to  write  £15  -f  J  jj  +  j^^  of  -^^.  And  in 
order  to  express  this  decimally,  we  would  have  to  reduce  tlie 
compound  fraction  to  a  simple  one,  and  then  the  vulgar  fractions 
to  decimals  by  last  article. 

French  Coins. 

The  franc  is  the  principal  unit  of  the  new  French  system  of 
coins,  its  divisions  being  the  decime  and  renflme.    The  Napoleon 


DECIMAL    SYSTEM    OP   WEIGHTS,    AC.  117 

contains  20  francs.  The  sou,  or  piece  of  5  centimes,  is  still  re- 
tained, but  all  calculations  are  made  with  the  franc  and  its  deci- 
mal divisions. 

TABLE. 

The  Franc  is  divided  into 10  decimes. 

The  Decime 10  centimes. 

Thus,  we  would  write  545  francs,  8  decimes,  (16  sous),  and  4 
centimes,  545-84/r. 

We  will  now  explain  the  nomenclature  of  the  French  system 
of  weights  and  measures,  to  which  the  name  metrical  system  has 
been  given,  the  metre  being  the  principal  unit. 

98.  The  unit  of  length,  to  which  we  give  the  name  metrej  is 
the  ten  millionth  part  of  the  distance  from  the  pole  to  the  equa- 
tor, measured  on  the  meridian  of  Paris.  According  to  measure- 
ments made  and  verified  with  the  utmost  precision,  the  metre^ 
valued  in  old  French  feet  and  inches,  is  equal  to  3  feetj  0  inches, 
11-296  line,  to  within  less  than  y^L-  of  a  line,  or  equal  to 
39-3809171  of  our  inches.*  In  order  to  designate  measures 
smaller  or  larger  than  the  metre,  it  is  agreed  upon  to  employ  the 
following  prefixes,  (taken  from  the  Greek  and  Latin). 

Myria,  Kilo,  Hecto,  Deca,  Bed,  Centi,  Milli,  which  signify  ten 
thousand,  thousand,  hundred,  ten,  tenth  of,  hundredth  of,  thou- 
sandth of,  (the  multiples  being  indicated  by  the  Greek  prefix, 
the  submultiples  by  the  Latin).  These  prefixes  are  placed  before 
the  word  metre  ;  and  the  following  table  is  formed.  For  conve- 
nience of  comparison,  we  convert  the  divisions  and  subdivisions 
into  parts  of  our  inch. 


*  This  measurement  of  the  arc  of  the  meridian  was  made  under  the 
auspices  of  Arago  and  Biot.  Several  degrees,  measured  with  great  accu- 
racy, served  as  a  basis  for  the  calculation  of  the  length  of  the  whole 
meridian. 


Myriametre, 

or 

Milometre, 

(C 

Hectometre, 

Decametre, 
Metre, 

Decimetre, 

= 

Centimetre 

=; 

Millimetre 

= 

"118  •      DECIMAL   SYSTEM   OF   WEIGHTS,    &C. 

10,000  metres  =  393809-171  inches. 

1000  metres  =  39380-9171       "      • 

100  metres  =  3938-09171       " 

10  metres  =  393-809171       " 

principal  unit  =  39-3809171       " 

-^\  of  a  metre  =  3-93809171       " 

yi^  of  a  metre  =  0-393809171     " 

-j-o^^j^  of  a  metre  =  0-0393809171  " 

N.  B.  The  myriametre,  and  the  Mlometrey  are  the  itinerary 
measures  at  present  adopted  in  France.  The  myriametre  is  6-22 
miles. 


Measures  of  Superficies;  or.  Square  Measure. 

99.  The  natural  unit  of  surface  is  the  square  metre;  that  is,  a 
square  which  has  a  metre  for  its  side.  The  decimetre  squared, 
or  the  square  which  has  a  decimetre  for  its  side,  is  jj^  of  the 
metre  squared;  the  square  centimetre  is  j^j^^^,  and  so  on,  for 
the  rest.  The  square  decametre  is  equal  to  100  square  metres. 
This  measure  we  take  for  the  principal  unit  in  all  field  measures; 
and  this  unit  is  called  are.  The  multiples  and  subdivisions  of 
the  are  are  also  designated  by  the  aid  of  the  prefixes,  myria, 
hectOj  deci,  centi  ....     Thus, 

The  Myriare  =  10,000  ares  = 

Kilare =     1000  ares 

Hectare =       100     « 

Decare =         10     " 

Are      =  the  principal  unit  =  100  sq.  ms.  =  119-665  sq.  yds.  =  ^ 
acre,  about. 

Declare   —  ^^  of  an  are. 

Centiare  =  -j^^  of  an  are. 

Milliare  =  jj^^-q  of  an  are. 


DECIMAL   SYSTEM  OF   WEIGHTS,   AC.  119 

N.  B.  The  myriarej  the  hectare,  are,  and  centiare,  are  the  only 
measures  used.     The  centiare  is  the  square  metre."^ 

Measures  of  Volume. 

100.  The  unit  of  volume  is  the  cuhic  metre  ;  that  is,  a  cube, 
(solid,  of  the  form  of  a  die),  which  has  a  metre  for  its  side.  The 
multiples  and  submultiples  of  the  cubic  metre  have  as  yet  re- 
ceived no  particular  names.  The  1000th  of  the  cubic  metre  is 
called  the  cubic  decimetre,  because  it  is  a  cube  with  a  decimetre 
for  its  side,  &c.,  for  the  cubic  centimetre  ....  When  the  mea- 
sures of  volume  are  applied  to  wood  for  burning,  or  to  materials 
of  building,  the  principal  unit  or  cubic  metre  is  called  stere.  We 
then  have  the  decastere,  or  measure  of  ten  steres.  The  stere  = 
3 5  "3 7 5  cuhic  feet. 

Measures  of  Capacity,  both  Dry  and  Liquid. 

101.  The  unit  of  capacity  is  the  cuhic  decimetre,  which  is 
called  litre. 

As  to  the  decimal  multiples  and  submultiples,  we  give  those 
which  are  chiefly  used. 

Hectolitre...,  =  100  litres. 

Decalitre =  .10  litres.  [cub.  in. 

Litre  =  principal  unit  =  1-057  U.  S.  qts.  =  61.074 

Decilitre =  t  o  ^^  ^  ^^t^Q- 

Centilitre =  jj^  of  a  litre. 

Weights. 

102.  The  unit  of  weight  is  the  weight  of  a  cuhic  centimetre 
of  distilled  water,  at  the  temperature  of  maximum  density,  viz., 
39-5°  Fahrenheit.  The  name  given  to  this  unit  \b gramme.  The 
gramme  is  equal  to  0-002204737  pounds  avoirdupois. 

*  A  partial  decimal  square  measure  has  been  introduced  among  sur- 
veyors in  the  United  States.  The  surveyor's  chain,  66  feet  in  length,  is 
divided  into  100  equal  links ;  and  we  have 

10,000  square  links     =  1  sq.  chain. 
10  square  chains  =  1  acre. 


120  DECIMAL  SYSTEM   OP  WEIGHTS,   &C. 

TABLE. 

lbs. 

The  Myriagramme  is =10,000  grammes=  22-04737 

Kilogramme =   1000  grammes  =  2-204737 

Hectogramme =      100  grammes=  0-2204737 

Decagramme  =        10  grammes  =  0  02204737 

Gramme =     principal  unit=  1  002204737 

Decigramme  =   -j-'^  of  a  gramme  =  0-0002204737 

Centigramme  =  j-ioOfagramme=  0-00002204737 

Milligramme  =    ^J^,  of  a  gra.  =0-000002204737 

N.  B.  The  half  kilogramme  is  about  equal  to  the  old  French 
pound,  nearly  equal  to  our  pound  avoirdupois. 

103.  Such  is  the  nomenclature  of  the  measures  which  compose 
the  metrical  system.  We  can  now  judge  of  the  advantages  which 
this  system  possesses  over  the  ordinary  measures. 

1st.  It  is  uniform  and  simple,  inasmuch  as  its  principal  units 
and  their  subdivisions  follow  the  law  of  the  decimal  system  of 
numeration. 

2d.  It  is  fixed,  invariable,  and  susceptible  of  being  adopted  in 
all  countries,  since  it  is  equally  adapted  to  any  climate  or  lati- 
tude. 

All  these  measures  have  for  their  base  one  primitive  measure, 
the  metre,  which  is  taken  from  the  dimensions  of  the  earth 
itself. 

We  will  dwell  but  little  upon  the  application  of  the  four  fun- 
damental operations  of  arithmetic  to  the  decimal  system  of  weights 
and  measures,  since  every  collection  of  principal  units  and  their 
subdivisions,  according  to  the  nomenclature,  can  be  expressed  by 
a  decimal  fraction  ;  and,  therefore,  these  operations  become  opera- 
tions upon  decimal  fractions,  considered  as  abstract  numbers.  For 
these  last  operations  we  have  already  established  fixed  rules. 
Nevertheless,  we  will  propose  some  questions  in  multiplication 
and  division,  because  they  will  afibrd  opportunity  for  some  im- 
portant remarks  upon  approximate  calculations. 


DECIMAL   SYSTEM   OF   WEIGHTS,    &C.  121 

Examples  under  the  different  tables  illustrating  the  above. 

1st.  —  56  kilometres,  25  decametres,  5  metres,  and  9  milli- 
metres, are  written,  56255-009  metres. 

2d.  —  25  hectares,  4  ares,  and  6  centiares,  are  written  2504-06 
ares. 

3d.  —  34  hectolitres,  and  6  centilitres,  are  written  340-06  litres, 

4th.  —  54  myriagrammes,  4  decagrammes,  7  decigrammes,  and 
3  milligrammes,  are  written  540040-703  grammes 

Multiplication. 

104.  Question  first.  —  Required,  the  price  of  35  metres,  429 
millimetres  of  a  certain  stuff,  one  metre  of  which  costs  $19  and 
76  cents. 

Here,  if  we  multiply  35-429  w.  by  $19-76,  we  will  obtain  a 
product  which,  expressed  in  dollars,  cents,  and  mills,  will  be  the 
price  required.  The  abstract  product  of  these  numbers  (93),  is 
700-07704;  then,  $700-07,  or,  more  exactly,  $700-08  is  the 
price  of  35-429  m.  Sometimes  the  fraction  of  the  metre  is  ex- 
pressed by  a  vulgar  fraction.  In  this  case,  the  operation  can  be 
performed  in  two  ways. 

Question  second.  —  What  is  the  price  of  23|  m.  of  a  piece  of 
stuff,  at  $8-25  cts.  per  metre  ? 

1st.  The  reduction  of  |  to  decimals,  gives  0-75;  the  question 
is  then  reduced  to  multiplying  8-25  by  23-75,  which  gives 
195-9375;  then,  $195-94  c^s.  is  the  price  of  the  23|  metres,  to 
within  less  than  ^  cent  (93). 

2d.  We  could  also  operate  as  follows : 
8-25 
23| 


24-75 
1650 


189-75 
i  =  4-125 
\  =  2-0625 

195-9375 


11 


122 


DECIMAL   SYSTEM   OF   WEIGHTS,    &G 


In  this  operation,  after  forming  the  product  of  ^he  two  entire 
parts,  we  have  added  the  two  partial  products,  and  placed  the 
point  where  it  properly  belongs,,  in  order  to  avoid  all  error.in  the 
final  result.  We  have  then  multiplied  8-25  by  |  (A  +  |)  by 
taking  first  the  half  of  8-25,  which  gives  4-125;  then  the  half 
of  this  half,  which  gives  2-0625.  Now,  taking  the  sum,  we  get 
195-8375,  as  by  the  first  method. 

This  last  method  of  proceeding  is  preferable,  when  the  vulgar 
fraction  cannot  be  converted  into  a  limited  number  of  decimal 
figures. 

Third  Question.  —  To  find  the  price  of  89  j^  metres,  supposing 
one  metre  to  cost  $47*19. 


1st  operation, 

4719 

424-71 
3775-2 

4199-91 
=  23-595 
=  11-7975 

=   7-8650 

42431675 

Then,  89  {^  metres  cost  M243-17,  to  within  less  than  one  cent. 

Otherwise,  commencing  by  converting  i^  into  decimals,  we 
find  (>-91666  .  .  .  .;  and  we  must  multiply  89-916666  ....  by 
47-19. 


89-91 
47-19 

89-916 
47-19 

89-9166 
4719 

80919 
8991 
62937 
35964 

809244 
89916 
629412 
359664 

8092494 
899166 
6294162 
3596664 

4242-8529 

424313604 

4243164354 

This  table  gives  three  distinct  operations.     1st,  with  two  de- 
cimal figures  of  the  multiplicand;  2d,  with  three;  3d,  with  four; 


DECIMAL   SYSTEM   OP   WEIGHTS,    &C.  123 

and  we  see  it  is  the  last  only  wliich  gives  the  approximation  to 
within  less  than  one  cejit. 

The  difficulty  here  is  to  know  how  many  of  the  decimal  figures 
of  the  multiplicand  we  must  take,  in  order  to  be  assured  that  we 
have  the  degree  of  approximation  required  by  the  nature  of  the 
question ;  while  by  the  first  method  we  obtain  a  complete  result, 
of  which  we  can,  according  to  choice,  neglect  more  or  less  of  the 
decimal  figures. 

N.  B.  We  could  also  reduce  89|-J  to  a  single  fraction ;  then 
multiply  47 '19  by  this  fraction;  an  operation  longer  than  the 
first  method  which  we  have  used. 

Division. 

105.  Question  Fourth. — A  piece  of  land  containing  23  hec- 
tares, 9  aresj  25  centiares,  (23  A.,  0925  c),  loas  bought  for 
$83,719-25.     Required  the  value  of  the  hectare? 

We  must  here  divide  83719-25  by  23-0925;  and  the  quotient, 
valued  in  dollars  and  cents,  will  represent  the  price  per  hectare. 

We  obtain,  by  simple  division  of  decimals,  $3625-38. 

Question  Fifth.  —  28^|  kihgrammes,  of  a  certain  material, 
cost  $519-35.      What  is  the  price  per  kilogramme  f 

Here  we  may  use  two  methods.  1st,  Reduce  28^|  (o  a  single 
fractional  number,  giving  ^^^^  Then  multiply  519-35  by  ^J, 
inverted,  (Art. -59);  we  thus  find  for  result,  18-038. 

2d.  We  convert  ^|  to  decimals,  which  gives  0-79166  .  .  .  .; 
then  we  divide  519-35  by  28-79,  taking  only  two  decimal  places 
of  the  divisor;  we  obtain  thus,  18-039.  Then,  $18-04  is  the 
price  per  kilogramme  of  the  stufi". 

These  examples  suffice  to  show  how  we  must  proceed  in  the 
multiplication  and  division  of  denominate  numbers  of  the  decimal 
systems,  and  to  show  how  much  simpler  these  operations  are  than 
in  the  ordinary  systems  of  compound  numbers. 


124  DECIMAL   SYSTEM   OF   WEIGHTS,    &C. 

We  will  add  here,  as  belonging  properly  to  tlic  preceding 
theories,  some  notions  upon  the  different  divisions  of  the  circle 
and  thermometer. 

106.  0/  the  two  divisions  of  the  Circle.  —  The  circumference 
of  a  circle  is  defined  in  geometry  a  recutiant  line,  all  the  points 
of  which  are  equally  distant  from  a  point  within,  called  the  centre. 
In  all  the  scientific  works  in  this  country,  the  circumference  is 
divided  into  360  equal  parts,  called  degrees  (°) ;  each  degree  into 
60  equal  parts,  called  miimte.s  (');  each  minute  into  60  equal 
parts,  called  seconds  (").  This  is  called  the  sexagesimal  division. 
When  the  French  reformed  their  system  of  weights  and  measures, 
they  adopted  also  a  centesimal  division  of  the  circumference  of 
the  circle,  the  use  of  which  is  becoming  very  general  among  the 
scientific  men  of  Europe.  In  this  new  centesimal  system,  the 
circumference  is  divided  into  400  equal  parts,  called  degrees  (°) ; 
each  degree  into  100  equal  parts,  called  minutes  (') ,'  each  minute 
into  100  parts,  called  seconds  (");  each  second  into  100  equal 
parts,  called  thirds  ('''),  &c. 

Example  of  Sexagesimal  Division.  — 45  degrees,  38  minutes, 
25  seconds,  are  written  45°  38'  25". 

Example  of  Centesimal  Division.  —  28  degrees,  56  minutes, 
and  23  seconds,  are  written  28-5623°,  in  the  decimal  form.  In 
order  to  reduce  the  divisions  of  the  sexagesimal  system  to  a  com- 
pound number  of  the  centesimal,  we  observe  that  the  quarter  of 
the  circumference,  called  a  quadrant,  is  in  one  system  90°,  and 
the  other  100°.  Then,  1°  sexagesimal  =  ^-^^^  or  y>  of  a  degree 
centesimal,  and  vice  versa;  1°  centesimal  =  f^  of  a  degree  sexa- 
gesimal. 

We  are  thus  led  to  the  two  following  rules : 

1st.  To  convert  a  compound  number  sexagesimal  to  a  com- 
pound centesimal.  Reduce,  first,  to  a  fractional  number  of  de- 
grees (JQ')]  then  multiply  this  number  by  ^-^j  and  convert  the 
result  into  decimals.  The  entire  part  will  express  the  centesimal 
degrees;  the  decimal  part,  divided  into  periods  of  two  figures 
each,  the  minutes,  seconds,  &c. 

2d.  Reciprocally,  to  convert  a  compound  centesimal  number 


AC.  12. J 

into  a  compound  sexagesimal.  Subtract  from  the  yiccn  number , 
expressed  in  decimal  form^  j'^  of  this  number,  (o?-  simpli/  take 
y^Q  of  if).  The  entire  part  of  the  result  will  represent  the  num- 
ber of  sexagesimal  degrees.  The  decimal  part  we  convert  into 
minutes  and  seconds  by  the  known  rules  for  converting  fractions 
of  a  higher  denomination  into  units  of  a  lower. 

Examples.  —  1st.  Convert  34°  69'  17"  sexagesimal,  into  de- 
grees, minutes,  and  seconds,  centesimal. — 34°  59'  17",  converted 
to  seconds,  give  125957",  or  ^*§|J^'''  of  a  degree;  this,  multi- 
plied by  Y?  g^v^s  ^lilB^-  Finally,  the  division  of  125957  by 
3240,  gives  38 -875617 or  38°  87'  56"  17'"  centesimal. 

Reciprocally,  2d.  —  To  convert  38° -875617  centesimal,  into 
degrees,  minutes,  and  seconds,  sexagesimal. 
38-8756170 
-j-V  3-8875617 


34-9880553 
60 

59-283318 
60 


16  99908     or,  34°  69'  17". 
Op  the  Principal  Divisions  of  the  Thermometer. 

107.  The  thermometers  mostly  used  on  the  continent  of  Eu- 
rope are,  the  thermometer  of  Reaumur,  and  the  Centigrade.  In 
England  and  the  United  States,  the  use  of  Fahrenheit's  thermo- 
meter is  almost  universal.  These  all  differ  in  their  scales  of  sub- 
division only.  In  Reaumur's,  the  interval  between  the  freezing 
and  boiling  points  of  water  is  divided  into  80  equal  parts,  called 
degrees  of  Reaumur;  in  the  Centigrade,  this  same  interval  is 
divided  into  100  parts,  called  centesimal  degrees.  It  follows,  that 
each  degree  of  Reaumur's  is  equal  to  "^^^^  or  |,  of  the  Centigrade 
degree;  and,  reciprocally,  each  Centigrade  degree  is  equal  to  | 
of  the  degree  of  Reaumur.  Moreover,  the  fractions  of  the  de- 
gree are  expressed  generally  in  both  by  decimal  fractions.  Thus, 
it  is  a  very  simple  matter  to  transform  one  into  the  other. 
11* 


126  DECIMAL    SYSTEM    OF    WEIGHTS,    &C. 

1st.  In  order  to  convert  a  decimal  number  of  de2;rees  of  R&iu* 
mur  into  Centigrade  degrees,  we  add  to  the  number  one-fourth 
of  itself.     The  result  of  the  addition  is  the  number  sought. 

2d.  In  order  to  convert  a  decimal,  number  of  centesimal  de- 
grees into  degrees  of  Reaumur,  subtract  from  the  given  number 
one-fifth  of  itself,  and  you  have  the  number  sought. 

Thus,  for  example : 

39°-4716  R.  ==  39-4716  +  9-8679  =  49°-3395  0. 
Reciprocally, 

49°-3395  C-  =  49-3395  —  98679  =  39°-4716  C. 

In  Fahrenheit's  thermometer,  the  freezing  point  of  water  is 
32°,  instead  of  0°,  and  the  interval  between  that  and  the  boiling 
point  (212°)  is  180°.  Then,  the  degree  of  Fahrenheit  is  }§§ 
=  ig,  or  i  of  the  degree  Centigrade;  and,  reciprocally,  the  de- 
gree Centigrade  is  |,  or  |§,  of  the  degree  of  Fahrenheit.  In  the 
actual  reduction  from  one  of  these  scales  to  the  other,  we  must 
always  keep  account  of  the  different  start  point,  both  for  negative 
and  positive  temperatures.     Thus, 

1st.  To  convert  a  decimal  number  of  degrees  Fahrenheit  (-f ) 
into  centesimal  degrees,  we  must  first  subtract  32° ;  then  remove 
the  decimal  point  one  place  farther  to  the  right,  and  divide  by 
18,  (or  multiply  by  5  and  divide  by  9). 

2d.  To  convert  a  decimal  number  of  degrees  Centigrade  into 
degrees  of  Fahrenheit,  remove  the  decimal  point  one  place  to  the 
left,  and  multiply  by  18,  (or  multiply  by  9,  and  divide  by  5) ; 
then  add  32°  to  the  result. 

Example  1st.  To  convert  56° -259  Fahrenheit  into  Centigrade 
degrees. 
56°-259— 32°=24°-259....24°-259xig=24_2^59=i3o.477c, 

2d.  To  convert  13°-48  C.  to  degrees  Fahrenheit. 
13-48  X  18=1-348  X  18=24-259  ....  24-259-f  32°  =  56°-259  F. 


DECIMAL   SYSTEM   OF   WEIGHTS,    AC.  127 

The  rules  for  the  conversion  of  the (minus)  degrees,  and 

also  for  conversion  of  Fahrenheit  into  Reaumur,  are  too  obvious 
to  discuss  them  farther. 

108.  General  Conclusion.  —  This  first  part  of  our  work 
includes  all  which  constitutes  elementary  arithmetic,  the  princi- 
pal object  of  which  is  the  exposition  and  development  of  the 
methods  to  be  followed,  in  order  to  perform  upon  numbers  all 
possible  operations.  These  operations  are  to  the  number  of  four 
fundamental  ones,  addition,  suhtraction,  multiplication,  and 
division.  All  the  others,  such  as  the  reduction  of  fractions  to 
the  same  denominator,  to  their  simplest  form,  the  conversion  of 
a  vulgar  fraction  into  a  decimal,  &c.,  are  nothing  more  than  com- 
binations of  those  which  we  have  just  given. 

There  are  two  other  species  of  operation,  or  rather  two  parti- 
cular cases  of  the  last  two  fundamental  operations,  of  which  we 
have  not  spoken ,  because,  in  order  to  be  developed  in  a  complete 
manner,  they  require  some  knowledge  of  algebra.  These  are  the 
formation  of  powers,  and  the  extraction  of  roots  of  numbers. 
The  powers  of  a  number  are  the  products  which  arise  from  the 
continued  multiplication  of  a  number  by  itself.  Thus,  4x4x4 
X  4  X  4  =  the  5th  power  of  4.  The  formation  of  powers  is  evi- 
dently then  a  particular  case  of  multiplication.  The  roots  of  a 
number  are  those  numbers  whose  continued  products,  each  by 
itself,  will  produce  the  given  number.  Then,  the  extraction  of 
roots  proposes  the  solution  of  the  problem —  Given  a  mimher,  to 
find  the  tioo  equal  factors  which  form  it,  or  the  three  equal 
factors,  &c. ;  evidently  a  particular  case  of  division.  We  will 
not  discuss  these,  because  they  are  fully  treated  in  all  of  the  good 
text-books  on  algebra. 

In  the  next  chapter,  we  propose  to  consider  numbers  in  a 
general  manner,  independently  of  every  system  of  numeration, 
and  to  develop  the  properties  belonging  to  any  given  system. 
This  will  be  in  some  sort  Arithmetic  Generalized. 


SECOND    PART. 


CHAPTER  V. 

GENERAL  Pl^OPEETIES  OF  NUMBERS. 

109.  Introduction.  —  Before  going  farther  into  tlie  science 
of  numbers,  and  in  order  to  investigate  their  properties  with 
more  facility,  we  must  borrow  from  algebra  some  of  its  materials, 
such  as  letters  and  signs  (some  of  which  we  have  used  already), 
by  the  aid  of  which  we  indicate,  in  a  general  and  abridged  man- 
ner, the  operations  and  the  reasoning  which  the  resolution  of  a 
question  requires. 

1st.  The  letters,  which  we  employ  instead  of  iSgures,  in  order 
to  represent  numbers.  Their  use  affords  at  once  a  mode  of  writing, 
more  concise  and  more  general  than  that  of  figures. 

2d.  The  sign  +  plus  (already  used),  to  indicate  the  addition 
of  two  or  more  numbers. 

3d.  The  sign  minus  —  (already  used),  to  indicate  the  subtrac- 
tion of  one  number  from  another. 

4th.  The  sign  of  multiplication  is  X ,  or  a  point,  which  we 
place  between  the  two  numbers,  read  multiplied  hy.  Thus,  aXh, 
or,  a.  hj  mean  a  multiplied  by  h. 

N.  B.  We  have  already  used  both  these  signs.  Now,  when 
the  numbers,  the  multiplication  of  which  we  wish  to  indicate, 

(128) 


GENERAL   PROPERTIES   OF   NUMBERS.  129 

are  expressed  by  letters,  then  this  multiplication  will  be  indicated 
also  by  simply  writing  one  of  the  letters  after  the  other,  with  no 
sign  between ;  thus,  ah  signifies  a  multiplied  by  h.  But  this 
method  cannot  be  employed  when  the  numbers  are  indicated  by 
figures ;  for,  if  we  wrote  the  product  of  5  by  6,  56,  this  notation 
would  be  confounded  with  fifty-six.  In  the  case  of  figures,  then, 
X  ,  or  some  such  sign  between  the  numbers,  is  necessary.  An- 
other sign  of  multiplication  is  the  parenthesis  (  ). 

5th.  The  sign  of  division,  either  a  bar  ( — ^),  as  already  used  in 
vulgar  fractions,  or  a  bar  with  two  points,  thus  (-^-),  or  simply 
two  points.     Thus,  \4  ==  24  --  6  =  24  :  6  =  24  divided  by  6 

6th.  The  Coefficient  is  the  sign  which  we  employ,  when  a 
number  denoted  by  a  letter  is  to  be  added  to  itself  several  times. 
Thus,  instead  of  writing  a-\-a-\-a-\-a-\-a,  which  represents 
the  number  a  added  to  itself  four  times,  we  write  5a.  We  say, 
then,  the  coefficient  is  the  number  written  on  the  left  of  another 
numher,  denoted  hy  one  or  more  letters,  to  show  how  many  times 
this  number  is  taken,  or  the  number  of  times  plus  one  it  is  added 
to  itself. 

7th.  The  exponent  is  the  sign  which  we  employ,  when  a  num- 
ber denoted  by  a  letter  is  multiplied  several  times  by  itself. 
Thus,  instead  of  writing  aXaXaXaxa,  or  aaaaa,  we 
write  simply  a^ ,  which  signifies  that  a  is  multiplied  4  times  by 
itself. 

The  exponent  is  then  a  number  written  to  the  right,  and  a  little 
above  another  number,  or  letter  expressing  a  number,  showing  the 
number  of  times  plus  one  that  this  number  or  representative  letter 
is  multiplied  by  itself 

8th.  The  sign  which  expresses  that  two  numbers  are  equal, 
already  used  (=),  read  is  equal  to,  or  simply  equals. 

The  axioms  applied  in  the  operati<.ns  on  equations  we  annex 
to  this  definition,  viz  :  If  equals  be  added  to,  subtracted  from, 
multiplied,  or  divided  by  equals,  the  results  will  be  equal. 

These  preliminaries  being  established,  we  will  take  up  again 
some  of  the  subjects  of  which  we  have  treated  in  the  first  part, 
in  order  to  investigate  them  more  thoroughly.     We  will  arrive 


130  GENERAL   PROPERTIES    OP   NUMBERS. 

thus  at  new  properties,  and  at  means  of  simplifying  -or  modifying 
the  methods  in  the  different  operations  of  arithmetic. 

[In  order  to  give  some  idea  of  the  use  of  these  different  signs, 
and  of  the  simplicity  of  the  algebraic  language,  we  will  make  a 
few  applications. 

Let  us  suppose,  first,  that  we  wish  to  express  that  a  number, 
represented  by  a,  is  to  be  multiplied  3  times  by  itself;  that  the 
product  thus  resulting  is  to  be  multiplied  3  times  successively  by 
h;  and,  finally,  the  new  product  is  to  be  multiplied  twice  by  c; 
we  will  simply-  write  a*iV. 

If  we  wish  to  express  that  it  is  necessary  to  add  this  last  result 
6  times  to  itself,  or  multiply  it  by  7,  we  write  Ta^tV. 

In  the  same  manner,  6a®6*  is  the  abridged  expression  of  6 
times  the  product  of  the  5th  power  of  a  by  the  second  power 
of  h. 

3  a — 55  is  the  abridged  expression  of  the  difference  between 
the  triple  of  a  and  the  quintuple  of  h. 

'±0^ — 3a6-j-45*  is  the  abridged  expression  of  the  double  of  the 
square  of  a,  diminished  by  the  triple  product  of  a  and  i,  and 
augmented  by  four  times  the  square  of  h. 

Let  us  now  see  how  we  can  effect,  upon  quantities  expressed 
algebraically,  the  fundamental  operations  of  arithmetic.  We  will 
limit  ourselves  to  the  most  simple  cases — those  to  which  we  will 
have  to  refer  in  the  latter  part  of  this  treatise. 

Addition.  —  In  order  to  add  two  numbers,  a  and  5,  we  write 
simply  a-f  6.  In  the  same  manner,  a-\-h-\-c  indicates  the  addi- 
tion of  the  numbers  a^  h,  c :  that  results  from  the  notation  we 
have  established.  In  the  same  manner,  a  —  6  and  c -J- r7 — /, 
added  together,  form  the  single  quantity,  a  —  h-\-c  -\-  d — f.  If 
we  had  to  add  a  —  h  and  h  —  r,  we  would  write  a  —  h  -{-h  —  c. 
But  as,  on  the  one  hand,  h  is  added,  and  on  the  other  subtracted, 
it  follows  that  these  two  operations  balance  each  other,  and  the 
expression  is  reduced  to  a  —  c. 


GENERAL   PROPERTIES    OF   NUMBERS.  131 

Subtraction.  —  In  order  to  subtract  h  from  a,  we  write  a  —  h. 
In  the  same  manner,  if  we  wish  to  subtract  c  from  a  —  6,  we 
write  a  —  h  —  c. 

Let  it  be  required  to  subtract  the  expression  e  —  d  from  the 
expression  a  —  h.  We  can  first  indicate  the  subtraction  thus: 
a  —  h  —  (c  —  c7).  But  if  we  wish  to  reduce  the  result  to  a  single 
expression,  we  must  reason  as  follows : 

If  we  had  to  subtract  c  alone  from  a  —  6,  the  result  would  be 
a  —  h  —  c.  Now,  as  it  is  not  o,  but  c  diminished  by  J,  which  is 
to  be  subtracted,  the  result,  a  —  h  —  c,  is  too  small  by  the  num- 
ber of  units  in  d ;  thus,  the  result  will  be  brought  back  to  its 
just  value  by  adding  d  to  a  —  h  —  c,  or  writing  a  —  h  —  c-{-d. 

That  is  to  say,  in  order  to  subtract  one  algebraic  expression 
from  another,  we  must  write  the  one  to  be  subtracted  with  the 
signs  of  all  its  terms  changed,  after  the  other  ;  thus  forming  one 
single  expression. 

We  find  by  this  rule,  and  analogous  reasoning, 
3a— (26— 3c)  =  3a— 25  +  3c. 
5a— 46  — (6c/— /-f  ^7)  =  5a— 46— 6c7  +f—g. 

Multiplication.  —  Eequired  to  multiply  a*  by  h^. 

We  write  a^  x  6',  or  simply  a^J?. 

But  if  we  have  a^  to  be  multiplied  by  a',  we  observe  that  the 
number  a,  being  5  times  a  factor  in  the  multiplicand,  and  3  times 
a  factor  in  the  multiplier,  ought  to  be  5  +  3,  or  8  times  a  factor  in 
the  product.  Thus,  we  have  a^Xa^^=a^;  that  is  to  say,  when 
the  same  letter  enters  into  both  factors  of  the  multiplication,  we 
lorite  it  once  in  the  product y  and  give  it  for  exponent  the  sum  of 
its  exponents  in  the  two  factors. 

Required,  now,  to  multiply  a  —  6  by  c. 

We  can  first  indicate  the  product  in  this  manner  (a  —  h)  c. 
But,  if  wc  wish  actually  to  perform  the  operation,  we  remark, 
that  to  multiply  a  —  6  by  c  (27),  is  to  multiply  c  by  a  —  h  ;  that 
is,  to  take  c  a  —  b  times,  or  as  many  times  as  there  are  units  in 
a,  diminished  by  the  units  in  b.  If,  then,  we  multiply  first  c  by 
a,  which  gives  ac  or  ca,  the  product  is  too  great  by  the  product 


132  GENERAL   PROPERTIES   OP   NUMBERS. 

of  c  by  h,  or  he.  Thus,  we  must  subtract  he  from  ac,  and  we 
obtain  ac  —  he  for  the  required  product,  (a  —  h)c  =  ac  —  he. 

Required,  again,  to  multiply  a  —  5  by  c  —  d. 

The  product  can  first  be  indicated  thus  :  (a  —  h)  (c  —  d). 

But,  in  order  to  obtain  a  single  expression,  we  commence  by 
multiplying  a  —  h  hj  c,  which  gives  ac  —  he;  and  we  observe, 
then,  that  it  is  not  by  c  alone  that  we  have  to  multiply  a  —  6, 
but  by  c  diminished  by  d. 

Thus,  the  product  ae  —  6c  is  too  large  by  the  product  of  a  —  b 
by  a;  that  is,  by  ad  —  hd.  Then,  in  order  to  reduce  the  pro- 
duct to  its  just  value,  we  must  subtract  ad  —  5c?  from  ac  —  be; 
which  gives,  by  the  rule  of  subtraction,  ac  —  he  —  ad  -\-  hd. 

Examining  this  product,  we  deduce  the  following  rule : 

In  order  to  effect  the  multiplication  of  two  algebraic  expressionSj. 
multiply  successively  each  term  of  the  multiplicand  by  each  term, 
of  the  multiplier  ;  observing,  that  if  two  terms  of  the  multiplicand 
and  multiplier  are  affected  with  the  same  sign,  their  product  is 
affected  with  the  sign  -j-  (^plus") ;  hut  if  they  are  affected  with 
different  signs,  their  product  is  affected  loith  the  sign  —  (minus). 

Division.  —  We  will  consider  only  a  single  case  of  this  opera- 
tion, in  which  the  two  terms  of  the  division  contain  the  same 
letters. 

Required  to  divide  a?  by  a^. 

We  can  first  indicate  the  quotient  in  this  manner : g,  or 

CT^-f-  a'.  But  a'  is  the  product  of  which  a^  and  the  quotient  are 
the  two  factors ;  hence,  the  exponent,  7,  of  the  dividend,  ought 
to  be  equal  to  the  sum  of  the  exponent  of  the  factor  known,  3, 
and  of  the  unknown  exponent  of  the  quotient;  then,  recipro- 
cally, this  last  is  equal  to  the  difference  between  the  exponent  of 
the  dividend  and  the  exponent  of  the  divisor  ;  that  is,  to  7  —  3 
or  4. 

Thus,  -r,  =  a\  -j-==ab  .  .  .  .  &c. 

'  a-*  a^b 

Such  are  the  general  notions  of  algebra,  of  which  we  will  have 

to  make  use  in  the  fifth  and  following  chapters.] 


GENERAL   PROPERTIES   OF   NUMBERS.  133 


Theory  op  Different  Systems  op  Numeration. 

110.  We  have  seen  (Art.  5),  how,  by  the  aid  of  ten  characters 
or  figures  we  can  represent  all  numbers,  setting  out  with  the 
conventional  principle,  jthat  every  figure  placed  on  the  left  of 
another,  expresses  units  ten  times  greater  than  those  of  the  first 
figure.  We  now  propose  to  show  that  we  can  write  all  numbers 
with  more  or  less  than  ten  characters,  provided  we  do  not  use  less 
than  two,  (zero,  0,  being  always  one  of  these  characters). 

We  call,  in  general,  the  number  of  figures  employed,  the  base 
of  the  system.  The  system  in  which  two  figures  are  used,  viz : 
(10),  is  the  binary  system,  and  2  is  the  base.  The  ternary 
system,  of  which  3  is  the  base,  makes  use  of  8  figures,  1,  2,  0 ; 
the  quaternary  has  four  figures,  1,  2,  3,  0;  the  quinary,  five, 
1,  2,3,  4,  0;  &c.,  &c. 

The  base  may  be  greater  than  ten ;  we  must  then  have  recourse 
to  additional  characters.  Thus,  in  the  system  of  which  twelve 
is  the  base,  the  duodenary  or  dvodecimal,  we  will  have  to  use 
two  new  signs,  a  and  jS,  to  express  ten  and  eleven  numbers  less 
than  the  base. 

In  every  system  analogous  to  the  decimal  system,  the  conven- 
tional principle  holds  that  every  figure  placed  on  the  left  of  an- 
otlierj  expresses  units  as  many  times  greater  than  those  of  the 
first  figure  as  there  are  units  in  the  base  of  the  system.  Thus,  in 
the  binary  system,  each  figure  acquires  a  value  twofold  greater 
for  each  place  that  it  is  removed  to  the  left.  In  the  ternary 
system,  they  increase  in  a  threefold  ratio ;  and,  in  general,  in  a 
system  of  which  b  is  the  base,  a  figure  goes  on  increasing  in  a 
b-f old  proportion,  as  it  is  removed  one  or  more  places  to  the  left. 

When  a  number  is  written  in  a  system  whose  base  is  b,  the 
first  figure  on  the  right .  expresses  the  units  of  the  first  order ; 
the  figure  immediately  on  its  left  the  units  of  the  second  order j 
the  next  figure  on  the  left  the  units  of  the  third  order  ;  and  so 
on.  It  requires  b  units  of  the  first  order  to  make  one  of  the 
second ;  b  units  of  the  second  to  form  one  of  the  third,  &c. 


134  GENERAL   PROPERTIES   OF   NUMBERS. 

111.  We  pass,  now,  to  the  manner  of  expressing  in  figures 
any  entire  number,  whatever  be  the  system  which  we  adopt.  In 
order  to  fix  our  ideas,  we  will  consider  the  septenary  system, 
which  makes  use  of  the  seven  characters,  1,  2,  3,  4,  5,  6,  0. 
Adding  unity  to  six,  we  obtain  seven,  or  the  unit  of  the  second 
order;  which,  according  to  the  principle  enunciated  above,  can 
be  expressed  by  10 ;  since  the  0,  having  no  value  of  itself,  makes 
the  figure  1  at  its  left  express  one  unit  of  the  second  order,  or 
seven  simple  units.  Placing,  successively,  all  the  figures  of  the 
system  in  the  first  and  second  place,  we  will  evidently  form  all 
the  consecutive  numbers  comprised  between  10  =  seven  and  the 
number  expressed  by  66. 

For  example,  11,  12,  13,  14,  15,  16,  represent  the  numbers 
eight,  nine,  ten,  eleven,  twelve,  thirteen,  20  =  fourteen,  21  = 
fifteen,  &c. 

After  reaching  the  number  66,  if  we  add  to  it  a  new  unit, 
there  will  result  6  units  of  the  second  order,  plus  seven  units  of 
the  first  order;  that  is  to  say,  seven  units  of  the  second  order, 
or  a  single  one  of  the  third  order,  which  can  be  expressed  by 
100.  Placing,  successively,  in  the  first,  second,  and  third  places, 
the  difi'erent  figures  of  the  system,  we  will  form  all  the  consecu- 
tive numbers  comprised  between  100  and  the  number  expressed 
by  666.  Reasoning  on  this  last  number,  as  upon  66,  we  shall 
arrive  at  the  unit  of  the  fourth  order,  which  is  expressed  by 
1000 ',  then  we  obtain,  successively,  all  the  consecutive  numbers 
comprised  between  1000  and  the  number  expressed  by  6666; 
and  so  on,  to  infinity;  whence  we  see  that  all  possible  entire 
numbers  can  be  written  in  this  system.  The  same  reasoning 
applies  to  any  other  system.  Whatever  system  be  adopted,  the 
units  of  the  different  orders  are  respectively  represented  by  1,  10, 
100,  1000,  10000,  &c.,  as  in  the  decimal  system. 

112.  N.  B.  We  have  said  (110),  that  the  character  0  was  in- 
dispensable in  every  system  analogous  to  the  decimal  system; 
that  is  to  say,  in  a  system  where  the  relative  value  of  a  figure 
depends  upon  the  place  which  it  occupies  on  the  left  of  several 


GENERAL   PROPERTIES   OP   NUMBERS.  135 

others.     To  speak  rigorously,  we  could  do  without  it;  but  the 
•system  would  be  less  regular,  as  we  shall  see. 

Let  it  be  proposed,  for  example,  to  establish  the  ternary  sys- 
tem, using  the  three  significant  figures,  1,  2,  3.  The  first  three 
numbers  are  expressed  by  these  figures.  In  order  to  represent 
four,jjivej  and  six,  it  would  suffice  to  write  11,  12,  13.  In  order 
to  express  seven,  eighty  nine,  ten,  eleven,  twelve,  we  would  write 

21  22  23  31  32  33 

In  the  same  manner, 

111  112  113  121  122  123 

would  express 

thirteen,  fourteen,  fifteen,  sixteen,  seventeen,  eighteen. 

It  is  not  necessary  to  go  farther,  in  order  to  see  the  inconve- 
niences of  this  system.  Its  principal  fault  consists  in  this,  that 
units  of  the  same  order  are  expressed  in  a  different  manner. 
Thus,  in  13  and  23,  the  figure  3  expresses  a  unit  of  the  second 
order,  the  same  with  the  figures  1  and  2  on  its  left.  In  123, 
23  express  nine,  or  units  of  the  third  order,  the  same  with  the 
figure  1  to  the  left  of  them.  (The  same  process  might  be  applied 
to  the  decimal  system  as  is  here  applied  to  the  ternary,  by  using 
a  single  character  for  ten,  and  dropping  the  0.) 

In  making  use  of  0,  it  suffices  to  determine  the  number  of 
units  of  different  orders  which  enter  into  the  proposed  number, 
and  to  write,  one  after  the  other,  the  figures  which  express  these 
units. 

113.  The  perfect  adaptation  of  the  nomenclature  of  numbers, 
and  the  manner  of  writing  them  in  figures,  in  the  decimal  system, 
permits  us  to  write  them  easily  from  dictation  in  ordinary  lan- 
guage. The  same  thing  would  be  true  of  every  system  of 
numeration  which  had  a  special  nomenclature  appropriate  to  the 
system ;  in  other  words,  a  spoken  numeration  corresponding  to 
its  written  one.  But  other  systems  do  not  present  this  immediate 
*  connexion  with  the  nomenclature  now  in  use. 


136  GENERAL   PROPERTIES    OF   NUMBERS. 

Let  it  be  proposed,  for  example,  to  express  the  number,  three 
hundred  and  sixtj/'inne,  referred  to  tlie  decimal  system  in  the 
septenary  system.  It  is  difficult  to  see,  d  priori,  which  are  the 
figures  proper  to  express  units  of  the  first,  second,  third  .... 
order  which  it  contains. 

Now,  since  this  number,  written  in  figures  in  the  decimal  sys- 
tem, is  three,  six,  nine,  it  follows  that  the  question  above  depends 
on  the  following,  which  is  much  more  general : — A  number  being 
enunciated  in  ordinary  language,  or  icritten  in  the  decimal  system, 
required  to  express  this  same  number  in  the  system  whose  base 
is  b. 

In  order  to  resolve  it,  we  remark,  that  since  it  takes  b  units 
of  the  first  order  to  make  one  unit  of  the  second  order,  as  many 
times  as  the  proposed  number  contains  the  number  b,  so  many 
units  of  the  second  order  of  the  system,  whose  base  is  b,  will  it 
contain ;  that  is  to  say,  that  if  we  divide  this  number  by  b,  the 
quotient  will  express  units  of  the  second  order,  and  the  remain- 
der, which  will  necessarily  be  less  than  b,  will  express  the  units 
of  the  first  order  of  the  number  written  in  the  system  whose 
base  is  b. 

In  the  same  manner,  since  b  units  of  the  second  order  in  the 
system  whose  base  is  b,  form  one  unit  of  the  third  order  in  the 
same  system,  if  we  divide  the  quotient  which  expresses  units  of 
the  second  order  by  b,  the  new  quotient  which  we  thus  obtain 
shall  express  units  of  the  third  order,  and  the  remainder,  always 
less  than  b,  shall  represent  the  units  of  the  second  order  written 
in  the  system  whose  base  is  b,  and  so  on  for  the  rest. 

Whence  we  see,  that  in  order  to  pass  from  the  decimal  system 
to  the  system  whose  base  is  b,  we  must,  1st,  divide  the  given 
number  by  the  base  of  the  new  system  written  in  the  decimal  sys- 
tem, and  write  the  remainder  of  this  division  apart,  as  expressing 
the  units  of  the  first  order  in  the  new  system  ;  2d,  divide  the  quo- 
tient obtained  by  the  same  base,  and  write  tJie  second  remainder 
to  the  left  of  the  first,  as  expressing  the  units  of  the  second  order  ; 
3d,  divide  the  second  quotient  by  the  same  base,  and  write  tJie 
third  remainder  on  the  left  of  the  two  preceding,  because  it  ex- 


GENERAL   PROPERTIES   OF   NUMBERS.  137 

presses  units  of  the  third  order  ;  continue  this  series  of  operations 
until  we  arrive  at  a  quotient  smaller  than  the  base  of  the  new 
system  ;  this  last  quotient  expresses  the  units  of  the  highest  order, 
and  is  written  on  the  left  of  all  the  remainders  successively  ob- 
tained. 

Let  us  apply  this  rule  to  the  number,  369,  which  we  wish  to 
express  in  the  septenary  system. 

7  I  369 
7  I  52     (5  1st  rem. 
7  1"    (3  2d  rem. 

1     (0  3d  rem.  • 

Dividing  369  by  7,  we  obtain  for  quotient,  52,  with  remain- 
der, 5,  which  we  write  apart,  in  order  to  express  the  units  of  the 
Jirst  order  in  the  new  system. 

Dividing  52  by  7,  we  find  7  for  quotient,  and  3  for  remainder, 
which  we  write  to  the  left  of  5,  as  it  expresses  units  of  the  se- 
cond order. 

Dividing  7  by  7,  we  have  1  for  quotient,  and  0  for  remainder, 
which  indicates  that  there  are  no  units  of  the  third  order;  but 
we  write  a  0  to  take  the  place. 

Finally,  as  the  quotient  1  is  smaller  than  7,  it  expresses  the 
units  of  the  fourth  order,  and  the  number  in  the  septenary  sys- 
tem is  (1035). 

On  examining  this  operation,  we  shall  find  that  we  have  ob- 
tained the  three  hundred  and  forty  threes,  the  forty  nines,  the 
sevens,  and  the  units,  which  the  given  number,  369,  contains 
Hence,  we  might  also  proceed  by  the  following  rule  : 

Find,  by  inspection,  the  highest  denomination  of  the  new  sys- 
tem which  the  given  number  contains;  divide  the  given  number 
by  the  number  expressing  the  highest  order  of  units  written  in 
the  decimal  system.  Set  the  quotient  apart,  as  expressing  the 
highest  order  of  units  of  the  required  number  in  the  new  system. 
Divide  the  remainder  by  the  number  expressing  the  value  of  the 
'next  lower  order,  and  place  the  quotient  on  the  right  of  the  first 
12* 


138  GENERAL   PROPERTIES    OP   NUMBERS. 

one  to  express  the  units  of  the  next  highest  order-.  Divide  the 
remainder  hy  the  next  lower j  placing  the  quotient  on  the  right  of 
the  two  preceding,  &c.,  &c. 

Thus,  in  the  same  example,  369  to  be  converted  into  its  equi- 
valent number  in  the  septenary  system.  We  see  that  three  hun- 
dred and  forty-three  is  the  highest  order  of  unit  of  the  septenary 
system  which  it  contains. 

We  divide  by  343 ;  the  remainder,  26,       343)369 ("1 
by  49 ;  the  remainder,  by  7  :  the  last  re  343 

mainder,  5,  being  necessarily  less  than  4q\9fi7o 

seven,  the  base  of  the  system  expresses 
units.  ^0  we  write  the  quotients  from  7)26(3 

left  to  right,  commencing  with  the  first  5  j^gt  rem. 

obtained,  and  write  the  last  remainder 

on  the  right  of  the  last  quotient.  We  obtain,  as  before,  (1035). 
The  first  method  given  is,  however,  the  best,  especially  for  large 
numbers. 

We  find,  by  this  method,  the  number  5347  of  the  decimal 
system,  equal  to  (12343)  of  the  system  which  has  eight  for  its 
base. 

8  I  5347 


1  1  668 

(3  1st  rem. 

8  1  83 

(4  2d  rem. 

8  1  10 

(3  3d  rem. 

1 

(2  4th  rem. 

(12343) 

Remarh.  —  It  can  happen  that  the  base  of  the  new  system  is 
greater  than  ten,  the  base  of  the  decimal  system.  In  this  case, 
we  proceed  as  follows  :  —  Required,  for  example,  to  convert  the 
number  8423  of  the  decimal  system  into  its  equivalent  number 
in  the  duodenary  system.  The  figures  of  this  system  are  1,  2, 
3,  4,  5,  6,  7,  8,  9,  a,  j3,  0.  (The  two  Greek  letters,  a  and  /3, 
being  employed  to  designate  ten  and  eleven  in  the  new  system.) 


GENERAL  PROPERTIES   OP   NUMBERS.  139 

12  I  8423 

12  |"701  (p  1st  rem. 

12  I  58  (5  2d  rem. 

4  (o  3d  rem. 

The  base  twelve  being  expressed  (4a5i3)  by  12  in  the  decimal 
system,  we  divide  8423  by  12,  which  gives  701,  and  remainder, 
/3  =  11,  in  the  decimal  system.  We  write  this  ^  apart,  as  ex- 
pressing units  of  the  first  order.  Likewise,  in  the  third  division, 
we  obtain  for  a  remainder  ten,  which,  in  the  new  system,  is  ex- 
pressed by  a;  we  then  write  a  to  the  left  of  the  two  figures 
already  found.  We  obtain  thus  (4a5/3)  for  the  equivalent  of 
the  given  number  in  the  new  system. 

114.  Reciprocalli/j  a  number  being  written  in  a  system  whose 
base  is  b,  required  to  enunciate  it  in  the  spoken  numeration  of 
the  decimal  system  ;  that  is,  to  convert  it  into  its  equivalent  in 
that  system. 

In  general,  let  ...  .  hgfdca  be  a  number  expressed  in  the 
system  of  which  b  is  the  base;  a,  c,  d,f,  &c.,  expressing  units 
of  the  first,  second,  third  ....  order,  (and  not  being  an  indicated 
product),  as  in  (4°,  Art.  109).  It  results  from  the  fundamental 
principle  established  in  (110),  that  the  figure  denoted  by  c,  ex- 
presses units  b  times  as  great  as  the  same  figure  standing  alone 
would  express;  then,  its  relative  value  can  be  represented  by 
cxb,  or  simply  by  cb  (109).  In  the  same  manner,  the  figure  a 
expresses  units  b  times  as  great  as  those  of  the  figure  c :  hence, 
its  relative  value  is  equal  to  the  product  of  d  hy  bxb  or  b^,  and 
can  be  expressed  db^.  We  could  show,  in  like  manner,  that/6^ 
g¥,  hb^  ....  are  the  relative  values  of  the  other  figures.  Then, 
the  given  number  is  expressed  by 

a^cb-\-db^+/b^-}-gb'+hb'+ 

Giving  to  the  base  b  and  to  the  figures  a,  c,  d,  /,  particular 
values,  we  effect  all  the  operations  indicated  in  this  expression, 
and  we  shall  obtain  the  number  corresponding  to  the  particular 
data,  converted  into  the  decimal  system. 


140  GENERAL   PROPERTIES   OP   NUMBERS. 

Required,  for  example,  to  convert  the  number  4867,  written 
in  the  system  of  eiyht  figures,  back  into  the  decimal  system. 

This  number  can,  according  to  the  expression  above,  be  placed 
under  the  form 

7  +  6x8-1-3x8^4x8'. 

We  have  at  once,  then, 

7       =7 

6x8 =48 

3x8' =192 

4x8=^ =  2048 

2295 

Adding  these  numbers,  we  have  2295  for  the  value  of  (4867) 
in  the  decimal  system.  We  can  verify  the  accuracy  of  this  ope- 
ration by  the  rule  of  (113). 

8  I  2295 

8  I  286  (7  1st  rem. 

Sjii"  (6  2d  rem. 

~4  (3  3d  rem.     (4867). 

And,  reciprocally,  this  last  operation  can  be  verified  by  the 
preceding  one,  which  we  will  enumerate  generally  thus : 

Form,  first,  the  different  powers  of  the  hase,  h,  written  in  the 
decimal  system;  multiply  then  all  the  figures  of  the  number, 
written  likewise  in  the  decimal  system,  as  a,  c,  d,  f  g,  h,  re- 
spectively, hy  1,  h,  h^,  h^,  h*,  V".  Adding  the  partial  products, 
we  shall  have  the  number  required. 

Given,  for  example,  the  number  (4a5)3)  in  the  duodenary 
system,  to  be  converted  into  its  equivalent  in  the  decimal  system. 
Since  a  and  |3,  written  in  the  decimal  system,  are  10  and  11  re- 
spectively, this  number  can  be  placed  under  the  form 

11  +  5x12  +  10x122+4x12'. 


GENERAL   PROPERTIES   OF   NUMBERS.  141 

11  =        11 

5x12 =      60 

10x12^ =1440 

4xl2» =6912 

8423 
Then,  (4a5|3)  equals  8423,  written  in  the  decimal  system. 

115.  The  two  preceding  rules  lead  to  a  third,  more  general, 
which  has  for  its  object  to  convert  any  number  from  a  system 
whose  base  is  6,  into  its  equivalent  in  a  system  whose  base  is  c. 

Convert  the  number  front  the  system  h  to  the  decimal  system^  hy 
(114);  then  from  the  decimal  system  to  tlie  system  c,  by  (113). 

Required,  for  example,  to  convert  the  number  (23104)  of  the 
system  whose  base  is  5,  to  its  equivalent  in  the  duodenary  sys- 
tem. We  obtain,  first,  for  this  number,  transformed  into  the 
decimal  system,  1654 ;  then,  for  this  last,  transformed  into  the 
duodenary  system,  ()35a).  We  can  verify  this  operation  by 
making  the  transformations  in  an  inverse  order. 

N.  B.  The  above  transformation  from  the  quinary  to  the  duo- 
denai-y  system,  could  be  effected  directly,  without  the  intervention 
of  the  decimal  system,  by  performing  all  the  operations  required 
in  the  quinary  system ;  the  only  difficulty  of  this  mode  of  ope- 
rating being  the  want  of  agreement  between  the  written  numera- 
tion of  this  system  and  the  spoken  numeration,  so  universally 
in  use. 

124.  The  methods  of  performing  the  four  fundamental  opera- 
tions of  arithmetic,  upon  numbers  written  in  any  system  whatever, 
do  not  differ  from  those  which  have  been  established  for  the 
decimal  system.  We  must  only  recollect  the  law  which  exists 
between  the  units  of  different  orders,  in  order  to  be  able  to  con- 
vert the  units  of  any  order  into  units  of  the  order  next  higher 
or  next  lower. 

In  order  to  familiarize  beginners  with  the  different  systems  of 
numeration,  we  will  propose  an  example  of  each  of  the  four 
operations  in  the  duodenary  system. 


142  GENERAL  PROPERTIES    OF   NUMBERS. 

1st.  Required  to  add  3704a,  i32956,  27i3a5,  48a/3.. 

We  find  for  the  sum  of  the  simple  units  thirty-two  ;  3704a 

that  is  to  say,  2  twelves  and  8  units;  we  then  write  8  132956 

in  the  units  column,  and  carry  2  to  the  column  of  units  27f3a5 

of  the  second  order.     The  sum  of  the  units  contained  48a/3 

in  this  second  column  is  thirty-one,  or  2  units  of  the  n-  g^o 
third  order,  and  7  of  the  second ;  we  write  the  7,  and 

carry  the  2  to  the  next  column.     Operating  in  the  same  manner 
on  the  other  columns,  we  obtain  (I5a678)  for  result. 

2d.  Required  to  subtract  from  5a0046 
The  number,  47a68p 

121577 

As  we  cannot  subtract  jS  from  6,  we  borrow  one  unit  of  the 
second  order  from  the  4,  and  say,  |3  from  eighteen  leave  7.  Pass- 
ing to  the  next  subtraction,  as  we  cannot  subtract  8  from  3,  we 
borrow  one  unit  from  the  first  significant  figure  to  the  left.  As 
there  are  two  zeros  between,  we  say,  this  unit  thus  borrowed 
equals  twelve,  or  |3  +  one  of  the  next  lower,  which  one  equals  J3  -f 
one  of  the  next  lower,  which  one  equals  twelve  of  the  same  order 
with  the  3.     We  then  Subtract  8  from  fifteen,  giving  7. 

In  the  two  following  subtractions,  we  regard  the  zeros  as  re- 
placed by  |3,  and  continue  the  operation  to  the  end. 

3d.  Required  to  multiply  3407a 
by     5a68 

228528 
180360 
294664 
148332 


177608828 


We  premise  here  a  table  of  multiplication  as  far  as  the  figure  |3, 
the  highest  figure  of  the  system,  after  the  manner  of  the  table 
of  Pythagoras. 


177608828 

5a68 

l^a08   • 
3a082 

3407a 

4a968 
0000 

GENERAL  PROPERTIES   OP   NUMBERS.  143 

This  being  premised,  we  multiply  3407a  by  8,  and  say;  8 
times  a  make  eighty,  or  (68)  of  duodenary  system ;  we  write  the 
8  and  carry  the  6.  Then  8  times  7  make  fifty-six,  and  6  make 
sixty-two,  or  (52)  of  the  duodenary.  We  write  the  2  and  re- 
serve 5  for  the  next  column.  Continiiinu;  this  operation,  we 
obtain  for  a  partial  product,  228528.  As  to  the  products  of  the 
multiplicand  by  the  other  figures  of  the  multiplier,  the  same 
reasoning  applies,  and  we  use  the  same  processes  as  in  the  deci- 
mal system.     Summing  up  the  products,  we  obtain  177608828. 

4th.  Let  us  verify  this  operation  by  divi- 
sion. We  simply  divide  the  product  ob- 
tained, by  one  of  the  factors.  In  order  to 
obtain  the  number  of  units  of  the  highest 
order  in  the  quotient,  we  take  the  first  five 
figures  on  the  left  of  the  dividend,  and 
divide  17760  by  5a68.  For,  thus,  we  see 
that  17  contains  5  three  times,  with  a  remainder.  Multiplying  the 
divisor  by  3,  and  subtracting  the  product  from  the  first  partial 
dividend,  we  obtain  for  a  remainder,  l/3aO.  '  We  bring  down  8 
and  divide  l|3a08  by  5a68,  obtaining  4  for  quotient,  and  3a0  for 
remainder. 

AVhen  the  following  figure  8  is  brought  down,  the  new  divi- 
dend does  not  contain  the  divisor ;  we  then  place  0  in  the  quo- 
tient and  bring  down  2,  which  gives  3a082  for  the  next  partial 
dividend.  Proceeding  in  the  .same  manner  with  the  rest,  we 
obtain  3407a  for  the  required  quotient. 

We  can  now  see  how  we  can  pass  at  once  from  the  number 
(23104)  of  the  quinary  system,  to  its  equivalent  in  the  duode- 
nary (115).  We  must  divide  23104  by  22,  or  twelve  written  in 
the  quinary  system,  and  perform  the  division  in  that  system,  we 
would  thus  obtain  a  remainder  which  would  express  the  units  of 
the  first  order  in  i\\Q  duodenary  system,  and  a  quotient  which  we 
would  divide  again  by  22,  or  twelve  expressed  in  the  quinary 
system,  in  order  to  get  the  units  of  the  second  order,  &c.,  &c. 

117.    General  RemarJc.  —  The  duodecimal  system  ofiers  some 


144  GENERAL   PROPERTIES   OF   NUMBERS, 

advantages  over  the  decimal,  inasmucKas  its  base  twelve  coniBins 
a  greater  number  of  factors  than  ten.  For  twelve  is  divisible  by 
2,  ^,  4,  6 ;  while  the  only  factors  of  10  are  2  and  5. 

Nevertheless,  we  could  not  substitute  the  duodenary  system,  or 
any  other,  for  the  decimal,  without  replacing  the  ancient  nomen- 
clature by  a  new  one,  which  was  more  appropriate  to  the  system 
adopted,  that  is,  which  made  the  enunciation  of  written  numbers 
easier. 

We  shall  perceive,  moreover,  that  the  greater  part  of  the  pro- 
perties of  numbers  which  have  been  discovered  are  true,  what- 
ever be  the  system  of  numeration  which  we  adopt,  and  some, 
which  shall  seem  to  belong  to  the  decimal  system  in  particular, 
have  their  analogous  properties  in  the  other  systems.  The  em- 
ployment of  the  letters  of  the  alphabet  in  order  to  represent 
numbers,  is  well  calculated  to  make  the  generality  of  these  pro- 
perties appear,  as  they  can  express  numbers  enunciated  in  any 
system  of  numeration  whatever. 

Principles   of    Multiplication   and    Division.     Divisi- 
bility OF  Numbers. 

118.  We  have  already  demonstrated  (25)  and  (26).  1st. 
That  to  multiply  a  number  hy  the  product  of  several  factors,  is 
tlie  same  thinff  as  multiplying  the  number  successively  by  each 
one  of  the  factors. 

2d.  That  the  product  of  two  numbers  is  the  same  in  whatever 
order  we  effect  their  multiplication. 

Though  the  reasoning  were  developed  upon  particular  num- 
bers, they  are  not  the  less  rigorous  on  that  account;  and  in 
order  to  convince  ourselves,  it  suffices  to  go  through  it  again, 
denoting  the  numbers  by  the  letters  a,  h,  c,  &c. 

We  propose  ta  verify  only  the  accuracy  of  the  second  of  the 
above  propositions,  whatever  be  the  number  of  factors  to  be  mul- 
tiplied together.  We  commence  by  remarking,  that  if  we  had 
to  multiply  a  number  N  by  6,  and  then  to  multiply  the  product 
obtained  by  r,  it  will  amount  to  the  same  thing  to  multiply  N  first 


GENERAL   PROPERTIES   OF   NUMBERS.  145 

by  c,  and  then  the  product  by  b.  In  other  terms  (8),  in  a  mul- 
tiplication of  more  than  two  factors,  we  can  invert  the  order  of 
the  last  two  multiplications  without  changing  their  product j  or, 
NxJxc  =  Nxcx  &.* 

For  it  results  from  the  first  principle  above,  that  N  X  6  X  c  = 
N  X  6c  ;  but  in  virtue  of  the  second  principle,  we  have  he  =  ch  ; 
then,  Nx  J  X  c  =  N  xc6,  or,  NxiXc=  N  x  cx6.     Q.  E.  D. 

From  this  proposition,  and  the  proposition  that  the  product  of 
two  numbers  is  the  same  in  whatever  order  we  take  them,  it  is 
easy  to  deduce  the  same  proposition  for  three  numbers. 

Let  a,  h,  c,  be  the  numbers  proposed. . 

We  say  that  ahc  =  hac  =s  bca  =  cha  =  cab  =  acb. 

For  the  second  product  is  equal  to  the  first,  in  virtue  of  pro- 
position 2d ;  the  third  is  equal  to  the  second,  in  virtue  of  pro- 
position 3d ;  the  fourth  is  equal  to  the  third,  in  virtue  of  2d ; 
the  fifth  is  equal  to  the  fourth,  in  virtue  of  the  3d ;  finally,  the 
sixth  is  equal  to  the  fifth,  in  virtue  of  2d.  Then  all  the  pro- 
ducts are  equal.  From  this  demonstration  for  three  factors,  and 
from  the  incidental  proposition  (3),  we  deduce  with  the  same 
faculty  the  proposition  for  four  factors. 

Let  a,  6,  c,  d,  be  the  numbers  proposed. 

We  say  then,  that 

abed  =  bacd  =  bead  :=  cbad  =  acbd  =  cabd 

=:  abdc  = 

=  bcda  = 

=  cadb  = 

Firstly,  the  six  products  of  the  first  horizontal  line  are  equal 
to  each  other,  in  virtue  of  the  proposition  for  three  factors,  since 
they  result  from  the  multiplication  of  ahc,  bac,  &c.,  &c.,  by  the 
same  number,  d.  The  first  product  of  the  second  line  is  equal 
to  the  first  of  the  first  line  by  reason  of  (3);  as  to  the  other  pro- 
ducts of  this  line,  we  dispense  with  writing  them ;  they  can  be 

*  We  here  for  convenience  sake,  shall  give  diflFerent  significations  to 
N  X  ^  X  c  and  N  X  ^c,  regarding  the  last  as  the  product  performed  of 
b  and  c. 

13 


146  GENERAL   PROPERTIES    OF   NUMBERS. 

found  easily,  keeping  c  in  the  last  place  in  each ;  they  are  all 
eqnal  to  the  first  by  the  proposition  for  three  factors.  We  could 
thus  proceed  with  the  other  two  lines,  applying  alternately  the 
incidental  proposition  (3),  and  the  proposition  for  three  factors. 
We  thus  prove  the  proposition  for  all  possible  products  of  a,  h, 
c,  d,  since  we  cannot  form  more  than  6  products  terminated  by 
the  same  letter.  The  same  mode  of  demonstration  can  obviously 
be  easily  extended  to  any  number  of  factors; 

119.  The  demonstration  which  we  have  given  of  the  pre- 
ceding principle,  supposes  that  the  numbers  upon  which  we  are 
reasoning  are  entire  numbers  (Arts.  25  and  26) ;  but  if  we  re- 
flect a  little  upon  the  rules  established  for  the  multiplication  of 
fractions,  we  perceive  that  the  property  is  equally  applicable  to 
fractional  numbers.  Moreover,  this  proposition  completes  the 
demonstration  of  the  method  established  for  the  reduction  of 
fractions  to  a  common  denominator  given  in  the  chapter  on 
fractions. 

Divisihility  of  Numbers. 

120.  The  property  which  certain  numbers  possess  of  being 
exactly  divisible  by  others,  and  the  investigation  of  the  divisors 
of  a  number,  form  one  of  the  most  important  theories  of  arith- 
metic. This  theory  depends  upon  a  series  of  principles,  which 
we  proceed  now  to  develop  successively. 

We  will  first  repeat  some  preliminary  definitions  which  we 
have  already  given.  We  say  that  every  entire  number,  which 
divides  exactly  another  entire  number,  is  called  z.  factor,  divisor , 
or  suhmultiple  of  this  number,  and  this  last  is  called  a  multiple 
of  the  first.  Every  entire  number  which  has  no  other  divisor 
except  itself  and  unity,  is  called  an  absolute  prime  number,  or 
simply  a  prime  number.  Two  entire  numbers  are  prime  with 
each  other  when  they  have  no  other  common  divisor  besides 
unity,  which  is  a  divisor  of  every  number. 

It  follows  from  this,  that  a  prime  number  which  does  not 
exactly  divide  another  number,  is  prime  with  the  latter,  as  they 
can  have  no  common  divisor  greater  than  unity. 


GENERAL   PROPERTIES   OF   NUMBERS.  147 

121.  First  Principle.  —  Every  number,  P,  which  divides 
exactly/  one  of  the  factors  of  the  product  A  X  B,  divides  neces- 
sarily the  product ;  or,  what  amounts  to  the  same  thing,  every 
entire  number  which  divides  another  exactly,  divides  necessarily 
the  multiples  of  this  number. 

For,  let  Q  be  the  quotient  supposed  exact  of  the  division  of 
A  by  P;  we  have  then  A  =  P  X  Q,  whence,  multiplying  both 
sides  by  B,  A  X  B  =  P  X  Q  X  IB  =  P  X  QB ;  we  see  then  that 
P  is  a  factor  of  the  product  AB. 

122.  Second  Principle. — Every  number  which  divides  exactly 
the  product  of  two  factors,  and  which  is  prime  with  one  of  them, 
divides  necessarily  the  other  factor. 

Let  A  X  B  be  the  given  product,  P  the  number  which  divides 
this  product  exactly ;  we  say  that  if  P  is  prime  with  A,  it  will 
divide  B. 

For  A  and  B  being  by  hypothesis  prime  with  each  other,  if 
we  apply  to  them  the  rule  of  the  greatest  common  divisor,  we 
will  be  led  to  a  remainder  equal  to  1 ;  that  is  to  say,  denoting  by 
r,  r'j  r" ,  ....  1,  the  successive  remainders,  we  will  have  the 
series  of  numbers. 

APr,  /,  r",  ...  1,  A  being  greater  than  P;  or,  PAr, 
/,  /',...  1,  if  P  is  greater  than  A,  for  the  different  terms  of 
the  divisions  to  be  performed.  But,  suppose  that,  before  perform- 
ing the  operations,  we  commence  by  multiplying  A  and  P  by  B, 
there  will  result  the  new  series,  AxB,  PxB,  rxB,  /xB 
....  1  X  B.  Now,  all  these  terms  are  divisible  by  P,  since  P 
is  the  common  divisor  of  the  two  first  terms.  Then  1  X  B,  or  B 
is  divisible  by  P.     Q.  E.  D. 

N.  B.  It  is  important  to  remark  that  the  proposition  is  only 
true  when  P  is  prime  with  one  of  the  factors  of  the  product. 

For,  if  we  have,  for  example,  on  the  one  hand  28  X  15,  and  on 
the  other  12,  which  is  not  prime  with  either  of  the  two  factors 
of  the  product,  the  quotient  of  the  division  of  (28  X  15),  or  420 
by  12,  is  exact  and  equal  to  35,  though  12  divides  neither  28 


148  GENERAL   PROPERTIES   OP   NUMBERS. 

nor  15.     It  is  obvious  in  tliis  case,  that  the  two  factors  contain 
together  all  the  prime  factors  which  compose  the  divisor. 
Thus  we  have 

28 X  15 -^ 4x  7x3x  5  =  (4x3)x  7x5. -=12x7x5=12x35 

Consequence  of  Second  Principle. — Any  number  whatever y  P, 
j)rime  with  all  the  factors  except  one  of  a  product^  A  X  B  X  C 
.  .  .  .,  can  only  divide  the  product^  when  it  divides  exactly  the 
remaining  factor. 

This  is  too  obvious  for  discussion. 

123.  Third  Principle.  —  Every  prime  number,  P,  which 
divides  Exactly  the  product  of  two  factors,  divides  one  of  them 
necessarily. 

For,  suppose  that  P  does  not  divide  A,  it  is  necessarily  prime 
with  A  (120);  then  it  must  divide  B  (122). 
From  this  result  the  following  consequences. 

124.  1st.  If  a  prime  number,  P,  divides  the  product  A  X  B 

X  C  X of  any  number  of  factors,  it  divides  one  of  the 

factors  at  least. 

2d.  Every  prime  number  which  divides  the  powers,  K^,  A^, 
A*,  &G.,  of  any  number.  A,  divides  A  itself.     For  A^,  A'',  &c., 

being  equal  to  A  X  A,  A  X  A  X  A P,  can  only  divide 

these  diflferent  products  when  it  divides  one  of  the  factors. 

3d.  If  two  numbers,  A  and  B,  are  prime  with  each  other,  their 
powers,  A^  and  B^,  A^  and  B^,  &c.,  are  also  prime  with  each 
other.  For  any  number,  a,  which  is  the  common  divisor  of  A^ 
and  B*,  for  example,  must  divide  A  and  B,  which  is,  by  hypo- 
thesis, impossible. 

125.  4th.  Every  number,  P,  prime  with  each  one  of  the  fac- 
tors of  a  product,  A  X  B  X  C  X  .  .  .  .,  is  also  prime  with  the 
product.     For  suppose  that  a  prime  number,  d,  differing  from  1, 

can  divide  at  once  P  and  the  product  AxBxC ,  asc? 

ought  to  divide  one  of  the  factors  of  the  product,  P  would  not 
be  prime  with  this  factor,  which  is  contrary  to  the  hypothesis. 


GENERAL   PROPERTIES   OF   NUMBERS.  149 

126.  5th.  Wheyi  a  nvmhcr,  N,  lias  heen  formed  hy  the  multi- 
plication of  several  others.  A,  B,  C,  D, . . .  .,  this  inimher  can  have 
no  other  prime  factors  except  those  which  'already  enter  into  A, 
B,  C,  Dj  &c.  For  every  prime  number  which  divides  the  pro- 
duct, A  X  B  X  C  X  D,  and  does  not  divide  J),  must  divide 
A  X  B  X  C  (123) ;  in  the  same  manner,  every  prime  number 
which  divides  A  X  B  X  C,  and  does  not  divide  C,  must  divide 
A  X  B,  and,  consequently,  A  or  B.  Thus,  we  can  say  in  other 
terms,  a  number  heimj  formed  hy  the  multiplication  of  several 
others,  we  cannot  obtain  it  anew  by  multiplying  numbers  which 
contain  prime  factors  different  from  those  which  enter  into  the 
numbers  already  multiplied. 

127.  Fourth  Principle.  —  Every  number,  N,  divisible  by  two 

or  more  numbers,  d,  d',  d", ,  prime  with  each  other,  is 

divisible  by  their  product. 

For,  since  d  divides  N,  we  have  N  =  c?  x  g',  q  being  an  entire 
number;  but  by  hypothesis,  </'  also  divides  N,  then  it  divides 
d  X  q,  and,  since  d  and  d'  are  prime  with  each  other,  d'  must 
divide  q  exactly;  and  we  have  q  =  d'  x  q'y  where  q'  is  an  entire 
number.  Hence,  N  =  c?  X  d'  xq\  and  N  is  divisible  by  dx  d'. 
In  the  same  manner,  we  can  continue  and  show  that  N  is  divisi- 
ble hy  dx  d'  X  d",  and  so  on  for  the  rest. 

128.  Consequence. — lid,d',d!',  .  .  .  .,  numbers  prime  with 
each  other,  enter  as  factors  into  N  a  certain  number  of  times, 
each  denoted  by  n,  n',  n" ,  .  .  .,  the  number  N  is  exactly  divisible 
by  (/■  X  (/'"'  X  d"""",  and  by  all  the  numbers  which  we  can  ob- 
tain by  multiplying  two  and  two,  three  and  three ;  the  different 
powers  of  d,  d',  d",  .  .  .  .,  comprised  between  the  first  and  the 
n*",  n'^^,  &c  ,  respectively.  For,  d,  d',  d",  .  .  .  .,  being  prime 
with  each  other,  (f",  ^'"',  d"^",  ....  must  be  so  also;  then 
(127),  their  products,  two  and  two,  three  and  three,  must  be 
exact  divisors  of  N. 

This  principle  serves  as  a  basis  in  the  investigation  of  all  the 
divisors  of  a  number. 
13* 


150  GENERAL   PROPERTIES    OF    NUMBERS. 

It  is  useless  to  observe,  that  all  the  propositions  established 
thus  far  are  true  in  all  the  systems  of  numeration.    ' 

We  will  give  some  now  which  relate  particularly  to  the  deci- 
mal system. 

129.    Signs  of  the  clivisihility  of  one  rmmher  hy  others. 

There  are  certain  signs  or  characteristics  by  which  we  can 
often  tell  whether  a  number  is  or  is  not  divisible  by  others.  A 
knowledge  of  these  is  often  useful  in  practice. 

The  reasoning  by  which  we  will  establish  these  signs  or  cha- 
racteristics of  divisibility,  rests  upon  the  following  principle. 

Let  a  number,  A,  be  divided  into  two  parts,  B  and  C,  so  that 
we  have 

A  =  B  -f  C  (1). 

1st.  If  a  fourth  number,  D,  divide  exactly  the  two  parts,  B 
and  C,  it  divides  also  their  sum. 

2d.  If  the  number,  J),  divides  one  of  the  parts,  B,  without 
dividing  the  other,  C,  it  will  not  divide  A ;  and  the  reraainder 
of  the  division  of  A  by  T),  is  equal  to  that  which  the  division  of 
G  by  D  gives. 

Of  the  first  principle,  it  is  very  easy  to  give  a  general  demon- 
stration.    Divide  (1)  by  D,  and  we  have 

D      D  ^  D  ^  •* 

B         C 
Now  the  two  terms,  -p-  and— j  are,  by  hypothesis,  entire  num- 

bers ;  then  the  first  number,  — ,  must  also  be  an  entire  number. 

As  to  the  2d  principle,  it  is  clear  from  the  above  equality  (2), 
that  if  B  is  divisible  by  D,  and  C  is  not,  A  cannot  be,  for  we 
would  otherwise  have  a  fraction  equal  to  a  whole  number. 

Again,  B  being  divisible  by  D,  we  have  B  =  DQ  (Q  an  entire 
number)  -,  C  not  being  divisible  by  J),  we  have  C  =  DQ'  -|-  R. 

Then,  B  -f  C  =  A  =  DQ  +  DQ'  +  R,  or  A  =  D  (Q-fQ')  +  R. 
Whence  we  see  that  A  divided  by  D,  gives  for  quotient,  Q-f  Q', 


GENERAL   PROPERTIES   OF   NUMBERS.  151 

and  remainder,  R,  of  the  division  of  C  by  D.     These  principles 
have  been  obviously  assumed  under  the  head  of  division. 

130.  Properties  of  the  numbers  2,  5,  4,  25,  8,  125, 

Is^  Every  nvmber  terminated  hy  one  of  the  figures,  0,  2,  4, 
6,  8,  is  divisible  hy  2.  For  this  number  can  be  decomposed  into 
two  parts,  viz  :  the  part  to  the  left  of  the  simple  units,  and  the 
collection  of  simple  units.  (For  example,  38576  is  equal  to 
38570-1-6).  Now  the  first  part  being  terminated  by  0,  is  a  mul-. 
tiple  of  10 ;  and  10  we  know  is  divisible  by  2 ;  then  this  first 
part  is  divisible  by  2.  And  if  the  second  part  contains  0,  2,  4, 
6,  8,  units  exactly,  it  is  divisible  by  2.  Hence,  the  number  is 
divisible  by  2. 

If  the  number  is  terminated  by  one  of  the  figures,  1,  3,  5,  7, 
9,  it  is  not  divisible  by  2,  since  one  of  its  parts  is  divisible,  and 
the  other  is  not.  The  numbers  divisible  by  2,  are  even  num- 
bers, the  others,  odd  numbers.  The  expression  2n  (n  being  any 
entire  number),  embraces  all  the  even  numbers  j  the  expression 
2n  4- 1,  all  the  odd  numbers. 

2d.  Every  number  terminated  by  a  0  {zero)  or  5,  is  divisible 
hy  5.  The  same  demonstration  as  before,  for  2.  If  the  last 
figure  is  different  from  0  or  5,  the  number  is  not  divisible  by  5 ; 
and  the  remainder  of  the  division  of  this  number  by  5,  is  equal 
to  the  remainder  of  the  division  of  the  last  Jiyure  by  5  (129). 
Thus,  1327  divided  by  5,  gives  for  remainder,  2,  equal  to  the  re- 
mainder of  the  division  of  7  by  5.  In  the  same  manner,  34789 
and  71436  give  for  remainders^  4  and  1. 

3d.  Every  number  of  which  the  two  last  figures,  taJcen  with 
their  relative  value,  form  a  number  divisible  by  4  or  25,  is  itself 
divisible  hy  ^  or  2b.  For  this  number  can  be  decomposed  into 
two  parts,  the  part  on  the  left  of  the  tens,  and  the  part  composed 
of  tens  and  units.  (For  example,  3548  and  27875,  are  equal  to 
3500  +  48  and  27800  -f  75).  Now,  the  first  part  being  ter- 
minated by  two  zeros,  is  a  multiple  of  100 ;  100  is  divisible  by 
4  or  25,  since  100  is  equal  to  25  X  4 ;  then,  this  first  part  is  also 


152  GENERAL  PROPERTIES   OP   NUMBERS. 

divisible  by  4  or  25.  Hence,  if  the  second  part  is  divisible  by 
4  or  25,  then  the  whole  number  is.  Thus,  3548  is'^divisible  by 
4,  because  48  is  a  multiple  of  4;  27875  is  divisible  by  25, 
because  75  is  a  multiple  of  25.  But  13758  is  not  divisible  by 
4,  and  gives  for  remainder,  2,  equal  to  the  remainder  of  the 
division  of  58  by  4 ;  25659  is  not  divisible  by  25,  and  gives  for 
remainder,  9,  or  the  remainder  of  the  division  of  59  by  25. 
For  the  number  25,  the  numbers  terminating  in  00,  25,  50,  or 
•  75,  are  the  only  numbers  divisible  by  25. 

4th.  Every  numher,  the  three  last  figures  of  which  considered 
with  their  relative  values,  form  a  number  divisible  by  8  or  125,  is 
also  divisible  by  8  or  125. 

The  demonstration  of  this  is  analogous  to  the  preceding.  It 
is  founded  upon  the  fact,  that  1000  =  125  X  8. 

131.  Properties  of  the  numbers  3  and  9.  Every  number,  the 
sum  of  whose  figures  is  divisible  by  3  or  by  9,  is  itself  divisible 
by  3  or  by  9.  And  the  remainder  of  the  division  of  any  num- 
ber whatever  by  3  or  by  9,  is  the  same  as  the  remainder  of  the 
division  of  the  sum  of  its  figures  by  3  or  by  9. 

We  remark,  first,  that  a  number  which  is  composed  of  unity, 
followed  by  one  or  more  zeros,  is  equal  to  a  multiple  of  9,  in- 
creased by  1.  (For  example,  10  =  9  -f  1 ,  100  =  99  +  1, 1000  = 
999  +  1,  all  the  parts,  9,  99,  999,  being  divisible  by  9  and  by  3). 
It  follows  from  this,  that  every  number  formed  by  a  significant 
figure  followed  by  one  or  more  zeros,  is  itself  a  certain  multiple 
of  9,  augmented  by  this  significant  figure.  For  example, 
70  =  7  X  10  =  7  (9  +  1)  =:=  7  X  9  +  7,  80000  =  8  x  (1000) 
=  8  X  (9999  +  1)  =  8  X  9999  4-  8. 

This  established,  let  us  take  any  number  whatever,  6205473, 
for  example.     It  can  be  decomposed  in  the  following  manner : 

6000000  +  2000004-0-00004-5000+400  4-70  +  3. 

And,  according  to  what  has  just  been  said,  it  contains  two  prin- 

cipal  parts.     1st.  A  sum  of  several  multiples  of  9,  which  sum  is 

itself  a  multiple  of  9  (129).     2d.  The  sum  of  the  figures 

6  +  2  +  0+5  +  4+7  +  3. 


I     \/Nvfvr 
GENERAL   PR^ERTIES   OF    NUMBERS.  153 

In  other  words,  the  number  can  obviously  be  written 

6  X  999999  +  6  +  2  x  99999  +  2  +  0  +  5.  x  999  +  4  x  99 
+  4  +  7x9  +  7  +  3. 

The  first  part  is  divisible  by  9.  The  second  part  is  or  is  not. 
In  the  first  case,  the  number  proposed  is  divisible  by  9 ;  and  in 
the  second  case,  the  remainder  of  the  division  of  the  significant 
figures  by  9  is  necessarily  the  remainder  of  the  division  of  the 
whole  number  by  9.  The  demonstration  for  3  is  absolutely  the 
same.  We  must  make  this  observation,  however,  when  the 
number  is  divisible  by  9,  it  is  necessarily  so  by  3 ;  but  it  can  be 
divisible  by  3,  without  being  so  by  9. 

To  establish  the  proposition  generally, 

Let  gfdcha  be  the  given  number,  which  we  will  denote,  more- 
over, by  N ;  we  have,  according  to  the  fundamental  principle  of 
numeration,  N=a+105-f  102c  +  10='cZ+10y+10V+  .  .  .  .,  an 
equation  which  can  be  placed  under  the  form 

N=  I      +(10-1)&+  (10^-l>-f  (10^-iy+(10^-l)/+  I 
U  +6  +c  +d  +/       +    j 

by  adding  and  subtracting  6,  c,  d,  &c.,  from  the  last  number  at 
the  same  time. 

Now,  according  to  what  was  premised  above,  10 — 1,  10^ — 1, 
10^ — 1  ....  being  divisible  by  3  or  by  9,  the  first  horizontal 
line  is  composed  of  a  succession  of  numbers  divisible  by  3  or  by 
9.  Thus,  this  first  part  of  the  number,  N,  is  divisible  by  9. 
Then,  if  the  second  part,  which  is  nothing  more  than  the  sum  of 
the  figures  of  the  given  number,  is  divisible  by  3  or  by  9,  the 
number  itself  is  divisible  by  3  or  by  9 ;  and,  if  this  last  part  is 
not  divisible  by  3  or  by  9,  the  remainder  of  this  division  will 
necessarily  (129),  be  the  remainder  of  the  division  of  the  num- 
ber itself,  by  3  or  by  9. 

-N.  B.  In  practice,  instead  of  determining  the  sum  total  of  the 
figures,  in  order  to  divide  it  by  9,  we  subtract  9  from  the  partial 
sum  so  soon  as  it  exceeds  or  equals  9,  as  we  proceed  with  the 


154  GENERAL   PROPERTIES   OF   NUMBERS. 

summing  up,  and  continue  the  operation  to  the  last  figure. 
These  partial  subtractions  do  not  obviously  change  the  remainder, 
which  we  seek. 

Example.  —  Given,  the  number 

74683056743. 

"We  say,  7  and  4  make  11 ;  9  from  11  leave  2 ;  2  and  6  make 
8,  and  8  make  16 ;  9  from  16  leave  7 ;  7  and  3  make  10 ;  9  from 
10  leave  1 ;  1  and  0  and  5  make  6,  and  6  make  12 ;  9  from  12 
leave  3 ;  3  and  7  make  10 ;  9  from  10  leave  1 ;  1  and  4  and  3 
make  8.  Then,  8  is  the  remainder  of  the  division  of  the  num- 
ber by  9. 

132.  Property  of  the  number  11.  —  Every  number  is  divisible 
by  11,  when  the  difference  between  the  sum  of  the  figures  in  the 
odd  places  J  counting  from  the  right,  and  the  sum  of  the  figures  in 
the  even  places,  is  equal  to  0,  or  divisible  by  11. 

Before  demonstrating  this  property,  it  is  necessary  to  re- 
mark, 

1st.  That  every  power  of  10  of  an  even  degree  dimi^iished  by 
unity,  gives  a  result  divisible  by  eleven. 

For  this  result  is  necessarily  composed  of  an  even  number  of 
9's,  written  one  after  another.  Now,  each  division  of  two  figures, 
taken  separately,  forms  99,  or  9  X  11,  divisible  by  11 ;  then,  the 
numbers  themselves  are  divisible  by  11 ;  or,  in  general,  10^" — 1 
is  divisible  by  11,  (2n  expressing  the  even  numbers). 

2d.  Every  uneven  power  of  10,  augmented  by  unify,  gives  a 
result  divisible  by  11. 

For  a  power  of  an  even  degree  of  the  number  10  can  be  ex- 
pressed by  10^"+'  (130).  Now,  10=^"+'=102"x  10,  or  10^-+'= 
lO^"  X  10  +  10  — 10  =  10  (10 2^—1)  +  10;  adding  1  to  both 
members,  lO^^+'  +  l  =  10(10^"— 1)  +  11.  But,- according  to  (1) 
10^" — 1  is  divisible  by  11;  moreover,  11  is  divisible  by  itself. 
Hence,  10^"+' +  1  is  also  divisible  by  11. 


GENERAL   PROPERTIES   OP   NUMBERS.  155 

This  being  established,  let  ...  .  lujfdcha  be  the  given  number, 
which  we  will  call  N ;  we  have  N==a-flO^*  +  10\-+10'^i4-10y. . . ., 
an  equation  which  we  can  put  under  the  form 

1  -\-a         —h  +c  —d  +/.  ...         ) 

Now,  according  to  the  two  preceding  remarks,  the  first  line  is 
composed  of  numbers  essentially  divisible  by  11,  and  forms,  con- 
sequently, a  first  part,  which  is  divisible  by  11.  Then,  if  the 
second  part,  which  is  nothing  more  than  the  difference  between 
the  sum  a  +  c-f-f+h-f-  .  .  .  .  of  (he  figures  in  the  odd  places,  and 
the  sum  of  tJie  figures  b-4-d  +  g+  .  .  .  .  of  the  figures  in  the  even 
places,  is  divisible  by  11,  as  we  have  supposed;  the  number,  N, 
is  also  divisible  by  11.     Q.  E.  D. 

133.  When  the  difi'erence  between  the  sum  of  the  figures  in 
the  odd  places,  and  of  those  in  the  even  places,  is  neither  0  nor 
a  multiple  of  11,  the  number  itself  is  not  divisible  by  11,  since 
one  of  its  parts  is  divisible,  and  the  other  is  not.  But,  then, 
there  are  two  cases  to  be  considered  with  reference  to  the  manner 
of  obtaining  the  remainder  of  the  division. 

1st.  If  the  sum  of  the  figures  of  the  odd  places  is  greater  than 
the  second  sum^  the  difference  is  to  be  added  to  the  first  horizon- 
tal line  of  the  value  of  N.  Denoting  then  this  first  line  by  B, 
and  the  difference  to  be  added  by  C,  we  will  have,  N  =  B  -f  C ; 
and  if  C  is  not  divisible  by  11,  the  remainder  of  the  division  of 
C  %  11  will  he  the  same  as  that  which  toe  icould  obtain  hy 
dividing  N  hy  11  (129). 

2d.  If  on  the  contrary,  the  sum  of  the  figures  of  the  odd 
orders  is  less  than  that  of  the  figures  of  the  even  orders,  the  dif- 
ference will  have  to  be  subtracted  from  the  first  line,  and  we 
shall  have  N  =  B  —  C ;  C  designating  always  the  numerical 
value  of  the  difference. 

In  order  to  determine  in  this  case  the  remainder  of  the  divi- 
sion  of  N  by  11,  let  us  observe  that  we  have  B  =  11  x  Q, 


156  GENERAL   PROPERTIES   OP   NUMBERS. 

Q  being  an  entire  number,  and  C  =  11  x  Q'  ■hJR-'}  then,  N  = 
11  X  Q  —  11 X  Q'  —  R,  or,  subtracting  and  adding  11, 

N=llxQ— 11x0^—11+ 11— 11=11  (Q—Q'—l)  +  ll—R. 

Whence  we  see  in  this  case  the  remainder  of  the  division  of  N 
by  11,  is  equal,  tiot  to  the  remainder  R,  of  the  division  of  C  b?/ 
11,  but  to  the  difference  between  II  and  11. 

In  order  to  fix  these  ideas,  let  the  number  be  47356708. 
Adding  up  the  figures  in  the  odd  places,  we  obtain  (setting  out 
from  the  right),  27  ;  adding  uj)  the  figures  of  the  even  orders,  we 
obtain  13.  Now,  the  first  sura  is  greater  than  the  second.  Then, 
if  we  take  the  diff'erence,  which  gives  14,  the  remainder,  3,  of 
the  division  of  this  diff'erence  by  11,  is  equal  to  that  of  the  divi- 
sion of  the  number  itself.  But,  if  we  had  the  number 
370546345,  since  the  sum  of  the  figures  of  the  odd  orders  is  15, 
and  that  of  figures  of  the  even  orders,  22,  it  follows  that  if  we 
take  the  diff'erence  between  the  two  sums,  which  gives  7,  the  re- 
mainder of  the  division  of  the  number  itself  is  not  7,  but  11 — 7, 
or  4. 

134.  Verification  of  multi-plication  and  division,  by  the  pro- 
perties of  9  and  11. 

We  cannot  pass  over  a  simple  and  very  convenient  means  of 
verifying  the  multiplication  and  division  of  entire  numbers.  We 
enunciate  this  method  as  follows  : 

Add  the  figures  of  the  multiplicand ,  and  divide  the  "sum  by  9 ; 
add.  the  figures  of  the  multiplier ,  and  divide  this  sum  also  by  9. 
We  thus  obtain  two  remainders,  which  (131),  are  nothing  more 
than  the  remainders  of  the  division  of  these  numbers  by  9.  Mul- 
tiply these  two  remainders  together,,  and  divide  their  product  by 
9;  this  gives  a  third  remainder.  Finally,  add  the  figures  of 
the  product,  and  divide  the  sum  by  9.  We  obtain  thus  a  fourth 
remainder,  which  is  equal  to  the  third  when  the  multiplication 
has  been  accurate.  Let  the  two  numbers  be,  for  example,  5786 
and  475,  to  be  multiplied  one  by  the  other.  The  multiplication 
being  performed,  we  add  the  figures  of  the  multiplicand,  rejecting 
the  9s  by  partial  subtractions,  as  in  (131).     We  thus  obtain  8  for 


GENERAL   PEOPERTIES   OF   NUMBERS.  157 

the  first   remainder.      We   operate  in   the  5786     8    2 

same    manner    on    the    multiplier    which  475      7    2 

gives    7    for  remainder.     This  7  we  write  28930 

under  the  8,  as  in  table.     We  then  multiply  40502 

8  by  7,  giving  56,  which  we  divide  by  9,        23144 

giving  2  for  remainder  (or  we  can  say  5  and        

6  make  11,  and  9  fronj  11  leave  2).    Finally,        2748350 
we  operate  upon  the  product  as  upon  the  factors,  which  gives  2 
for  a  fourth  remainder.     This  being  equal  to  the  third,  we  con- 
clude that  the  operation  is  exact. 

In  order  to  establish  this  method  of  verification  by  9  in  a 
general  manner,  let  us  denote  by  A  and  B  the  two  factors,  by  Q, 
Q',  R,  and  R',  the  quotients  and  the  remainders  of  the  division 
of  the  multiplier  and  multiplicand  by  9 ;  we  have  the  following 
equations, 

A  =  9  X  Q  +  R, 

B  =  9  X  Q'  +  R'. 

Multiplying  these  two,  member  by  member,  we  obtain 

AB  =  9x9xQxQ'4-9xQ'xR+9xQxR'+RxR'. 

Now,  the  three  first  terms  of  the  second  number  of  this  new 
equation,  are  evidently  multiples  of  9;  then  (129),  the  re- 
mainder of  the  division  of  the  product  AB  by  9,  must  be  that 
which  the  division  of  Rx  R'  by  9  gives.  And  this  is  what  we 
wished  to  demonstrate.  If  one  of  the  two  factors  of  the  multi- 
plication is  divisible  by  9,  the  product  ought  to  be  so  also ;  it  is 
the  same  if  the  product  R  x  R'  is  divisible  by  9.  Or,  we  may 
express  it  thus :  if  one  of  the  first  remainders  is  Oj  the  third  must 
also  be  0.  Hence,  the  fourth  must  be  0.  Again,  when  the  first 
two  remainders  are  equal  to  3,  in  which  case  the  third  remainder 
is  equal  to  0.  Hence,  the  fourth  must  be  0.  As  to  the  verifi- 
cation of  division,  two  cases  can  occur;  either  there  will  be  a  re- 
mainder after  the  ordinary  operation  is  performed,  or  there  will 
be  none. 

1st.  If  there  is  no  remainder,  the  dividend  is  regarded  as  the 
product  exact  of  the  quotient  and  divisor ;  and  we  can  apply  the 
14 


158  GENERAL  PROPERTIES    OF   NUMBERS. 

preceding  nile  regarding  the  divisor  and  quotient,  as  the  two 
factors  of  a  multiplication. 

2d.  If  we  obtain  a  remainder,  we  commence  by  subtracting 
this  remainder  from  the  dividend.  The  result  of  this  subtrac- 
tion will  be  the  exact  product  of  the  quotient  and  divisor,  and 
we  operate  upon  these  three  as  before. 

N.  B.  The  verification  hy  9  is  liable  to  several  causes  of  error , 
of  which  the  following  are  the  principal. 

1st.  It  is  possible  that  either  in  the  partial  products  or  in  the 
total  product,  we  may  have  written  a  0  for  a  9,  or  reciprocally ; 
or,  in  the  one,  a  figure  too  small  or  too  great  by  a  certain  number 
of  units,  and  in  the  other,  a  figure  too  great  or  too  small  by  the 
same  number  of  units. 

2d.  I.t  is  possible,  also,  when  there  are  zeros  in  the  multi- 
plier, that  we  may  not  have  written  the  partial  products  far 
enough  to  the  left.  We  perceive  at  once,  in  these  difi'erent  cases, 
that  the  errors  committed  have  no  influence  upon  the  remainders 
of  the  division  by  9,  of  the  terms  of  the  operation  to  be  verified. 

The  verification  by  9  is  only  then,  properly  speaking,  a  half 
proof,  to  which  we  can  have  recourse  when  pressed  for  time )  it 
being  certain  when  the  third  and  fourth  remainders  are  not 
equal,  that  the  operation  is  incorrect.  But  if  they  are  equal, 
there  is  only  a  great  prohahility  that  the  product  is  the  required 
one. 

The  verification  by  11,  which  does  not  differ  from  that  by  9, 
except  in  the  manner  of  obtaining  the  remainder  of  the  division 
of  a  number  by  11,  is  preferable,  though  itself  subject  to  some 
errors;  but  these  errors  occur  much  less  often  than  in  the 
method  by  9.  These  verifications  can  be  applied  equally  to  the 
multiplication  and  division  of  decimal  fractions,  since  these  ope- 
rations are  performed  in  the  same  manner  as  in  whole  numbers. 

135.  There  exist  also,  characteristics  by  which  we  can  tell 
whether  a  number  is  divisible  by  the  prime  numbers,  7,  13,  17, 
.  .  .  . ;  but  the  rules  which  it  is  necessary  to  follow,  are  longer 
in  practice  than  the  division  of  the  number  by  7,  13 


GENERAL   PROPERTIES   OP   NUMBERS.  159 

These  questions  demand,  moreover,  a  greater  knowledge  of  alge- 
bra than  the  questions  heretofore  discussed. 

We  will,  however,  give  the  following  question  as  an  exercise 
for  the  pupil ;  to  determine  in  any  system  of  numeration  what- 
ever, whose  base  is  b,  what  numbers  enjoy  properties  analogous 
to  the  properties  of  9  and  11  in  the  decimal  system,  and  to  de- 
monstrate these  properties.  This  can  be  solved  very  readily  ac- 
cording to  the  principle,  that  in  every  system  of  numeration,  any 
power  whatever  of  the  base  can  be  expressed  by  unity,  followed 
by  as  many  zeros  as  there  are  units  in  the  exponent  of  the  power. 

136.  As  to  the  characteristics  of  the  divisibility  of  a  number 
by  the  multiples,  6,  12,  15,  18,  86,  45,  of  the  prime  numbers, 
2,  3,  5,  they  are  sufficiently  simple  to  find  a  place  here. 

1st.  An  even  number  is  divisible  by  6  or  18,  when  the  sum 
of  its  figures  is  divisible  by  3  or  9.  For  this  number  is  then 
divisible  by  2  and  3,  or  9 ;  now  2  and  3,  2  and  9,  are  prime 
with  each  other;  then  (127),  the  number  is  divisible  by  6 
or  18. 

2d.  A  number  is  divisible  by  12  or  36,  when  the  two  last 
figures  form  a  number  divisible  by  4,  the  sum  of  the  figures  of 
the  number  being  at  the  same  time  divisible  by  3  or  9.  For 
then,  &c. 

3d.  Finally,  a  number  is  divisible  by  15  or  45,  when  the  last 
figure  is  0  or  5,  and  in  addition  to  this,  the  sum  of  the  figures  is 
divisible  by  3  or  9. 

We  pass  now  to  the  method  of  finding  all  the  divisors  of  a 
number,  both  prime  and  multiple.  ' 

137.  We  will  divide  this  question  into  two  distinct  parts  : 
The  first  has  for  its  object  to  determine  all  the  prime  factors 

which  enter  into  any  given  number,  and  the  number  of  times 
that  each  prime  factor  enters. 

The  second  has  for  its  object  to  obtain  all  the  divisors,  prime 
or  multiple,  which  the  number  contains. 


2820 

2 

1410 

2 

705 

3 

235 

5 

47 

47 

1 

160  GENERAL   PROPERTIES   OF   NUMBERS. 

First   Part.  —  To  decompose  a  number  into  ^11  its  prime 
factors. 

Let  the  number  be  for  example,  2820. 


2820=22x3x5x47. 


We  draw  first  a  vertical  line,  to  the  left  of  which  we  place  the 
number,  the  divisors  to  be  written  to  the  right  of  the  same  line : 
2820  being  divisible  by  2,  which  we  write  opposite  it  on  the 
right  of  the  vertical  line.  We  perform  the  division  of  2820  by 
2,  and  write  the  quotient,  1410,  below  the  2820.  As  1410  is 
divisible  by  2,  we  place  this  second  divisor  below  the  first ;  then 
the  resulting  quotient,  705,  below  the  preceding,  and  we  have 

2820=2^x705. 

Now,  we  say,  that  the  search  for  the  prime  divisors  of  2820, 
other  than  2,  is  now  reduced  to  finding  the  prime  divisors  of 
705.  For,  1st.  Every  divisor  of  705  must  divide  its  multiple 
2^x705.  2d.  Eeciprocally,  every  prime  divisor  of  2820,  other 
than  2,  must  divide  705. 

We  are  then  to  operate  upon  705  as  upon  the  given  number; 
705  is  divisible  by  3 ;  we  write  this  new  divisor  under  the  pre- 
ceding; then  we  place  the  corresponding  quotient,  235,  under 
the  last  already  obtained,  and  from  this  results  the  new  equality 

2820=2^x3x235. 

235  not  being  divisible  by  3,  the  question  is  reduced  to  finding 
the  prime  divisors  of  235.     Now,  this  number  is  divisible  by  5, 
which  we  write  in  the  column  of  divisors.     The  quotient  of  235 
by  5,  47,  we  place  in  the  column  of  quotients. 
We  have  then  the  equation 

2820=2^x3x5x47. 


GENERAL   PROPERTIES   OF   NUMBERS.  161 

"We  arc  now  led  to  seek  the  prime  divisors  of  47.  But  47  is 
obviously  itself  prime ;  for  the  simplest  prime  number,  after  5, 
is  7 ;  and  7  will  not  divide  it.  Moreover,  7  X  7=49,  a  number 
greater  than  47 ;  whence  we  conclude  that  47  is  a  prime  number. 
Dividing  it  by  itself,  we  set  the  quotient,  1,  below  the  others. 
Here  the  operation  ceases,  and  we  have  2820=2^x3x5x47 
for  the  number  2820,  decomposed  into  its  prime  factors. 

138.  Important  Remark.  — Before  going  farther,  let  us  gene- 
ralize what  has  just  been  said,  in  order  to  prove  that  47  is  a 
prime  number  \  we  will  thus  establish  for  every  number  a  limit 
above  which  it  is  useless  to  go  in  the  search  for  its  prime 
divisors. 

Let  N  be  the  given  number,  and  suppose  that  we  have  tried 
in  vain,  as  divisors,  all  the  prime  numbers  up  to  ascertain  number, 
a,  the  corresponding  quotient  of  which  is  g-,  a  fractional  number 
less  than  a.  We  say,  that  the  trial  of  any  other  number  would 
be  useless,  and  that  N  is  a  prime  number 

For  we  have,  according  to  the  supposition,  N  =  axg'  {q  being 
fractional  and  less  than  a).  Now,  if  there  existed  a  number  a' 
greater  than  a,  which  could  exactly  divide  N,  we  would  have, 
denoting  the  quotient  by  g', 

N=a'X2'  {^  being  an  entire  number). 

Whence,  a  X  q  =  a'  x  ^.  Now,  a'  being  greater  than  a,  ^ 
must,  to  compensate,  be  less  than  q,  which  is  itself  less  than  a. 
Hence,  the  number  N  would  have  an  entire  divisor  less  than  a, 
which  is  contrary  to  our  hypothesis. 

Take,  for  example,  the  number  263.  No  one  of  the  prime 
numbers,  2,  3,  5,  11,  13,  will  divide  this  number.  But,  trying 
17,  we  find  a  fractional  quotient,  15  +  1%,  a  number  less  than 
17 ;  whence,  we  conclude,  that  263  is  a  prime  number. 

In  general,  the  limit  of  the  trials  in  the  search  for  the  prime 
divisors  of  a  number,  is  the  smallest  prime  number  which  gives  a 
fractional   number  less  than  this  number  taken  for  the  divisor. 
There  are  other  limits  which  we  will  not  investigate. 
14* 


162  GENERAL   PROPERTIES   OF   NUMBERS. 

Let  us  now  render  general  the  method  which  we'  have,  for  the 
sake  of  clearness,  commenced,  by  developing  upon  a  particular 
example. 

Let  a  be  the  smallest  prime  number,  commencing  with  2, 
which  divides  N.  We  divide  N  by  a,  the  quotient  by  a,  the 
second  quotient  by  a,  as  long  as  the  exact  division  is  possible. 
Calling  n  the  number  of  divisions  which  we  have  found  it  possi- 
ble to  perform,  we  have  the  equation  N=a"xN'  (N'  being 
entire).  We  pursue  the  same  course  of  reasoning  as  in  (137), 
to  show  that  the  question  is  now  reduced  to  operating  upon  N' 
in  the  same  manner ;  fcalling  h  the  simplest  prime  number  which 
divides  N',  and  jp  the  number  of  successive  divisions  which  can 
be  performed,  we  have  N  =  a"xZ>PxN",  (N"  being  entire), 
admitting  that  c  and  d  are  the  only  factors  of  N",  so  that  we 
have 

N''  =  c-i  X  N'",  and  N"'c?», 
we  obtain 

N  =  a"  X  Jp  X  ci  X  (7% 

and  the  number  N  is  thus  decomposed  into  its  prime  factors ; 
and  we  know,  too,  the  number  of  times  that  each  one  of  these 
factors  enters  into  it. 

It  results,  moreover,  from  the  general  proposition  (126),  that 
these  prime  factors,  raised  to  the  powers  denoted  hy  the  exponents, 
n,  p,  q,  s,  respectively,  form  the  only  system  of  prime  factors 
into  which  the  number,  N,  can  he  decomposed. 

140.  Second  Part. —  To  determine  all  the  divisors,  hoth  simple 
and  multiple,  of  any  number  whatever. 

From  the  same  form  under  which  we  have  just  represented 
the  number  N,  results  a  method  of  resolving  this  question.  Let 
us  write 

1,  a,  a^,  a^,  a'',  ....  a"  (n  +  1  terms). 
1,  h,  b\  b%  b\  .  .  .  .  b^  {p-^l  terms). 
1,  c,  c^,  c^,  c*,  .  .  .  .  c«  (2  +  1  terms). 
1,  d,  d",  d%  d\  .  .  .  .  d'  (s-\-l  terms). 


GENERAL   PROPERTIES   OP   NUMBERS.  163 

It  is  evident  that  we  would  obtain  all  the  divisois  of  N,  unity 
included,  by  multiplying  all  the  terms  of  the  first  line  by  all  of 
the  second,  then  all  the  terms  of  the  product,  by  all  the  terms 
of  the  third  line,  and,  finally,  all  the  terms  of«  the  new  product 
by  those  of  the  fourth  line,  since  the  different  terms  of  this  last 
product,  are  the  products  1  and  1,  2  and  2,  3  and  3  .  .  .,  of  a, 
hj  c,  .  .  .,  raised  to  powers  whose  degrees  do  not  exceed  n,^,  q, 
and  s.  Now,  the  number  of  this  last  product  is  (w-f-l)  X  (p+1) 
X  (g'  4-  1)  X  (s  4-  1).  From  this  we  deduce  the  following  rule, 
also,  for  the  total  number  of  divisions  of  any  number. 

Increase  hy  unity  the  exponents^  n,  p,  q,  s,  .  .  .  of  the  differ- 
ent prime  factors  which  enter  into  the  number,  N.  Then  multi- 
ply together  these  exponents,  thus  augmented  by  unity  ;  the  pro- 
duct expresses  the  total  number  of  divisors  of  N,  unity  being 
comprised  among  the  number. 

Let  N,  for  example,  be  equal  to 

2^  X  3^  X  5^  X  7  X  13^ 
The  expression  above  becomes,  in  this  case, 

4x3x6x2x  3,  or  432; 
thus,  the  number  N  has  432  divisors. 

141.  The  method  which  we  have  just  indicated  for  determin- 
ing all  the  divisors,  prime  and  multiple,  of  a  number,  being  not 
very  convenient  in  practice,  we  will  explain  upon  a  new  ex- 
ample, a  more  expeditious  process. 
1 


5880 

2 

2940 

2,4 

1470 

2,8 

735 

3,  8,  12,  24 

245 

5,  10,  20,  40 

15, 

49 

7,  14,  28,  56 

21, 

30,  60,  120 

42,  84,  168  I  35,  70,  140  |  280  | 
105,  210,  420,  840 
7,  49,  98,  196  |  147,  294,  588,  1176  |  245,  490 
980,  1960  I  735,  1470,  2940,  5880. 
1 
In  all,  4  X  2  X  2  X  3,  or  48  divisors. 


164  GENERAL  PROPERTIES   OP   NUMBERS.  ' , 

Explanation  of  the  Table, 

After  having  determined  the  prime  divisors  of  5880,  by  the 
method  of  (137), 'we  write  1  above  the  factor  2,  in  the  column 
of  divisors.  We  pass  to  the  second  divisor,  2,  by  which  we  mul- 
tiply the  preceding;  this  gives  the  new  divisor,  4,  which  we 
place  on  the  right  of  the  second  divisor.  Passing  to  the  third 
divisor,  2,  we  multiply  4  only  by  2,  and  place  the  product  on 
the  right  of  the  third  divisor.  Passing  to  the  divisor,  3,  we 
multiply  it  by  all  the  divisors  which  precede,  viz :  2,  4,  8 ; 
which  gives  the  new  divisors,  6,  24,  48,  which  we  place  on  the 
right  of  the  divisor  3. 

In  a  word,  when  we  descend  to  a  new  divisor,  we  multiply  all 
the  divisors  which  precede  by  this  divisor,  taking  care  not  to 
repeat,  however,  the  products  already  obtained.  It  is  certain 
that  the  products  to  which  this  mode  of  proceeding  leads,  com- 
prehend all  the  divisors  of  the  given  number ;  since  they  are  the 
combinations  of  the  factors,  2,  3,  5,  7,  raised  respectively  to 
powers  whose  exponents  do  not  exceed  3  for  2,  1  for  3,  1  for  5, 
and  2  for  7. 

142.  The  search  for  the  prime  factors  of  every  number,  is  one 
of  the  most  important  questions  of  arithmetic,  and  one  of  the 
most  useful  in  practice.  One  of  the  applications  we  have  seen 
in  finding  the  least  common  multiple  of  several  numbers.  We 
may  also  apply  it  in  finding  the  greatest  common  divisor  of  two 
numbers,  this  being  obviously  the  product  of  all  the  prime  fac- 
tors common  to  the  two  numbers.     Thus,  for  example  : 

We  find  for  the  prime  divisors  of  the  number  2150,  1  x  2  x 
5  X  5  X  43,  and  for  the  number  3612,  1  X  2  x  2  x  3x7x43. 
Hence,  the  G.  C.  D.  of  these  two  numbers  is  2  x  43=86.  The 
reasons  are  obvious  from  the  preceding  articles. 

Formation  of  a  Table  op  Prime  Numbers. 

143.  The  principles  which  we  have  established  concerning 
prime  numbers,  and  the  application  which  have  been  made  of 


GENERAL   PROPERTIES   OF   NUMBERS.  165 

them,  show  sufficiently  the  utility  of  a  table  of  this  sort  of  num- 
bers, as  extended  as  possible. 

There  are  several  tables  of  this  sort,  some  of  them  compie- 
hending  all  the  prime  numbers  from  1  to  3036000. 

To  give  some  idea  of  the  manner  in  which  such  tables  are 
made,  suppose  that  we  wished  to  form  a  table  of  prime  numbers 
from  1  to  1000. 

The  first  thousand  numbers  are  written  one  after  another  in 
the  most  convenient  form  possible ;  for  example,  in  ten  columns, 
containing  one  hundred  numbers  each. 

We  then  proceed  as  follows  : 

We  draw  lines  across,  1st,  all  the  even  numbers  except  2 ; 
2d,  all  the  multiples  of  3  except  3,  which  remain  after  the  first 
operation;  3d,  and,  in  the  same  manner,  the  multiples  of  5, 
other  than  5,  which  have  not  been  crossed  in  the  first  two  opera- 
tions. This  done,  we  can  affirm  that  all  the  numbers  which 
have  not  been  thus  marked,  from  1  to  7  X  7,  or  49,  are  prime 
numbers,  since  all  the  multiples  of  2,  3,  and  5,  as  well  as  the 
multiplies  of  7,  below  this  limit,  have  necessarily  been  marked ; 
and  we  have  thus  the  prime  numbers  from  1  to  47.  In  the  same 
manner,  if  we  mark  all  the  multiples  of  7,  from  49  up  to  121, 
or  11  X  11  (11  being  the  prime  number  which  comes  directly 
after  7),  we  are  then  certain  that  the  numbers  preceding  121, 
which  are  not  marked,  are  prime  numbers;  we  thus  obtain  all 
the  prime  numbers  from  47  to  113,  inclusive. 

Without  carrying  the  details  of  this  operation  any  farther,  it 
is  easy  to  see  that  we  are  thus  led.  to  suppress,  successively,  all 
the  multiples  not  yet  suppressed,  of  the  prime  numbers  already 
found,  11,  13,  17,  .  .  ,  until  we  arrive  at  the  number  997,  the 
last  one  remaining  of  the  first  thousand  numbers,  after  the  suppres- 
sion already  made  of  998,  999,  and  1000,  as  multiples  of  2  and  3. 
We  find,  thus,  the  succession  of  169  prime  numbers  comprised 
between  1  and  1000,  the  table  of  which  we  subjoin,  adding  the 
six  prime  numbers  which  follow  them. 


1G6 


GENERAL   PROPERTIES   OF   NUMBERS. 


Tahle  of  Prime  Numbers  from  1  to  1033. 


1 

97 

229 

379 

541 

691 

863 

2 

101 

233 

383 

547 

701 

877 

3 

103 

239 

389 

557 

709 

881 

5 

109 

241 

397 

563 

719 

883 

7 

113 

251 

401 

569 

727 

887 

11 

127 

257 

409 

571 

733 

907 

13 

131 

263 

419 

577 

739 

911 

17 

137 

269 

421 

587 

743 

919 

19 

139 

271 

431 

593 

751 

929 

23 

149 

277 

433 

599 

757 

937 

29 

151 

281 

439 

601 

761 

941 

31 

157 

283 

443 

607 

769 

947 

37 

163 

293 

449 

613 

773 

953 

41 

167 

307 

457 

617 

787 

967 

43 

173 

311 

461 

619 

797 

971 

47 

179 

313 

463 

631 

809 

977 

63 

181 

317 

467 

641 

811 

983 

59 

191 

331 

479 

643 

821 

991 

61 

193 

337 

487 

647 

823 

997 

67 

197 

347 

491 

653 

827 

1009 

71 

199 

349 

499 

659 

829 

1013 

73 

211 

353 

503 

661 

839 

1019 

79 

223 

359 

509 

673 

853 

1021 

83 

227 

367 

521 

677 

857 

1031 

89 

373 

523 

683 

859 

1033 

144.  Remark  upon  the  greatest  common  divisor. 

We  may  find  it  necessary  sometimes  to  find  the  greatest 
common  divisor  of  several  numbers.  For  this  we  give  the 
following  rule.  We  find  first  the  Gr.  C.  D.  of  two  of  the  num- 
bers, then  the  G.  C  D.  of  the  one  already  found  and  a  third 
number y  then  the  G.  C.  D.  of  this  last  common  divisor  and  a 
fourth  number 

Let  A,  B,  C,  E,  F,  .  .  .  be  the  given  numbers,  and  call  D 
the  G.  C.  D.  of  A  and  B,  D'  the  G.  C.  D.  of  C  and  D.  Then 
we  say  that  D'  is  the  G.  C.  D.  of  A,  B  and  C.  For  the  G.  C.  I). 
of  A,  B  and  C,  must  divide  D,  and  moreover  must  divide  C. 
Hence,  the  greatest  number  which  divides  both  C  and  D,  is  the 
greatest  common  divisor  of  A,  B  and  C,  and  D'  is  that  number. 
The  same  course  of  reasoning  will  apply  to  the  rest  of  the 
numbers. 


GENERAL  PROPERTIES   OP   NUMBERS.  167 

We  see  that  there  is  some  advantage  in  operating  first  upon 
the  two  simplest  numbers,  since  the  Gr.  C.  D.  sought  cannot  exceed 
that  which  exists  between  these  two  numbers. 

We  could  also  decompose  the  numbers  into  their  prime  factors, 
and  proceed  as  in  the  method  proposed  in  (142). 

145.  Remark  upon  tlie  least  common  multiples. 

We  have  already  given  a  method  of  finding  the  least 
common  multiple  of  several  numbers  in  the  chapter  on  vulgar 
fractions,  which  is  rendered  complete  by  the  method  of  obtaining 
ihQ  prime  factors  of  any  number  whatever j  given  in  (137,  138). 
We  give  here  another  method  founded  upon  the  preceding  theo- 
ries. We  consider,  first,  the  two  numbers,  A  and  B.  Denoting 
their  greatest  common  divisor  by  D,  and  by  q^  ((  the  quotients 
of  the  division  of  A  and  B  by  D,  we  have  the  two  equations, 
A=Dx  2",  B==D  X  ((;  q  and  ^  being  prime  with  each  other. 

Now,  we  say,  that  the  least  common  multiple  required  is 
equal  to 

D  X  ^  X  2'. 

For  this  product  is  obviously  a  multiple  of  A  and  B,  since  it  is 
divisible  by  D  X  g-  and  D  X  q';  it  remains  to  be  proved  that  it  is 
the  least  multiple  which  we  can  obtain. 

Let  us  call  Mamj  multiple  whatever  of  A  and  B.  In  order 
to  be  divisible  by  A  or  D  X  g',  M  must  contain  all  the  factors 
which  enter  into  each  one  of  the  numbers,  D  and  q;  for  the 
same  reason  it  must  contain  all  the  factors  of  each  of  the  numbers 
D  and  g' ;  and  since  q  and  q^  are  prime  with  each  other,  M  can- 
not be  less  than  J)  X  q  X  q'-  We  have  then  the  following  rule : 
Determine  the  Gr.  C.  D.  of  A  and  B ;  then  divide  A  and  B  by  the 
Gr.  C.  D. ;  multiply  the  Gr.  C.  D.  and  the  product  of  the  two  quotients ; 
this  gives  M  for  the  least  common  multiple  of  A  and  B.  Find 
the  least  common  multiple  in  the  same  manner  of  M  and  C. 
This  will  be  the  L.  C.  M.  for  A,  B,  C.  Operate  in  the  same  man- 
ner on  all  the  rest  of  the  numbers  in  succession.  The  method 
given  under  the  head  of  reducing  fractions  to  the  least  common 


168  GENERAL   PROPERTIES   OF   NUMBERS. 

denominator,  can  be  reduced  to  practice  thus,  (now  that  we  know 
the  method  of  jBnding  all  the  prime  factors  of  any  number). 

For  example,  take  the  numbers,  6,  9,  4,  14,  and  16. 

2)6,    9,    4,    14,    16 

2)3     9     2       7      8 

3)3     9     1       7      4 

13     17       4  2x 2x3x3 X7x4=least com.  mult. 

We  place  the  numbers  in  a  horizontal  line,  and  commence  with 
the  prime  number,  2,  as  a  divisor.  We  divide  all  those  numbers 
which  are  divisible  by  2,  and  bring  down  the  quotients,  together 
with  the  numbers  not  divisible.  We  proceed  in  the  same  manner 
with  the  quotients,  until  there  are  no  two  which  2  will  divide. 
We  then  divide  the  last  quotients  and  numbers  brought 
down  by  the  prime  number,  3,  and  continue  the  operation 
until  there  are  no  two  numbers  left  divisible  by  any  number 
greater  than  unity.  We  then  multiply  the  divisors  and  the 
numbers  thus  remaining  together  for  the  least  common  multiple. 
It  is  evident  that  we  thus  form  the  least  number  divisible  by  the 
given  numbers. 

Or  Periodical  Decimal  Fractions. 

146.  The  valuation  of  vulgar  fractions  by  decimals,  that  is  to 
say,  by  tenths,  hundredths  ....  of  the  principal  unit,  gives  rise 
to  singular  circumstances  which  merit  an  examination.  But,  be- 
fore entering  upon  the  discussion  of  them,  we  must  return  to  the 
method  for  converting  a  vulgar  fraction  into  a  decimal. 

We  have  seen  that,  in  order  to  effect  this  reduction,  we  must 

1st.  Annex  a  0  to  the  numerator',  and  divide  the  resultimj 
number  hy  the  denominator  ;  this  gives  the  tenths  of  the  quotient 
and  a  remainder.  2d.  Write  a  new  0  on  the  right  of  the  re- 
mainder,  and  divide  hy  the  denominator ,  obtaining  thus  the 
hundredths  of  the  quotient.  We  continue  this  operation  until  we 
have  reached  the  degree  of  approximation  required.     This  pro- 


GENERAL  PROPERTIES   OP   NUMBERS.  169 

cess  is  evidently  the  same  as  multiplying  the  numerator  hy  unity ^ 
followed  hy  as  many  zeros  as  we  wish  decimal  figures  in  the  re- 
sult; then  dividing  the  result  hy  the  denominator,  and  pointing 
off  in  the  quotient  the  numher  of  decimal  figures  required. 

147.  This  enables  us  to  demonstrate  the  two  following  pro- 
perties : 

1st.  Every  vulgar  fraction  whose  denominator  does  not  contain 
any  prime  factors  other  than  2  and  5,  is  reducible  to  a  limited 
numher  of  decimal  figures  ;  that  is  to  say,  after  a  certain  num- 
ber of  operations,  we  must  arrive  at  a  remainder  equal  to  0 ;  in 
which  case  the  decimal  fraction  obtained  expresses  the  exact 
value  of  the  given  fraction.  Besides,  if  the  fraction  is  reduced 
to  its  simplest  form,  the  total  riumher  of  operations  to  he  per- 
formed in  order  to  reduce  it  to  its  equivalent  decimal  is  always 
equal  to  the  greatest  of  the  two  exponents  of  2  and  5,  which  enter 
into  the  denominator. 


Thus,  the  fractions 


7       13       11         317 
5J    3Tf?    4TJJ    T3S0> 


which  can  be  placed  under  the  forms 

2?      52'     2\b      2.5*' 

are  reducible  to  a  limited  number  of  decimal  figures.  The  first 
gives  rise  to  three  operations,  the  second  to  two,  the  third  to 
three,  and  the  fourth  to  4. 

We  find,  in  fact,  for  their  values, 

0-875;  0-52;  0-275;  0-2536. 

In  order  to  prove  this  property  generally,  we  remark,  that 
10,  100,  1000  ....  being  equal  to  2x  5,  22x  b\  2'x  5^  •  •  ., 
if,  in  order  to  effect  the  reduction  to  a  decimal  fraction,  we  mul- 
tiply the  numerator  by  10,  100,  1000  .  .  .  .,  the  resulting  product 
will  necessarily  be  divisible  by  2  X  5,  2^  X  5^  .... ;  then,  in 
15 


170  GENERAL   PROPERTIES   OF   NUMBERS. 

multiplying  this  numerator  by  unity,  followed  by  as  many  zeros 
as  there  are  units  in  the  greatest  of  the  exponents  of  2  and  5, 
"which  the  denominator  contains,  the  resulting  product  will  ne- 
cessarily be  a  multiple  of  this  denominator. 

Then,  the  number  of  operations  to  be  performed  is  equal  to 
the  greatest  of  the  two  exponents  of  2  and  6,  which  enter  the 
denominator  of  the  given  vulgar  fraction. 

148.  Every  irreducible  vulgar  fraction^  whose  denominator 
contains  one  or  more  prime  factors  different  from  2  and  6, 
gives  rise  to  an  infinite  number  of  decimal  figures.  Moreover j 
the  decimal  fraction  resulting  from  it  is  periodical ;  that  is  to 
say,  after  a  certain  number  of  operations,  the  same  decimal 
figures  recur  again. 

For  the  multiplication  of  the  numerator  by  10,  100,  1000,  can 
only  cause  the  introduction  of  the  two  factors,  2  and  5,  raised  to 
certain  powers ;  thus,  the  prime  factor  which  we  suppose  to  be 
in  the  denominator,  and  not  in  the  numerator,  will  not  be  in  the 
latter,  after  this  multiplication  by  10,  100,  ....  Then,  whatever 
number  of  zeros  we  add,  we  shall  never  obtain  a  product  exactly 
divisible  by  the  denominator;  thus,  the  operations  can  be  carried 
on  to  infinity. 

We  say,  moreover,  that  the  decimal  fraction  will  be  periodical. 
For,  as  each  remainder  is  always  less  than  the  divisor,  it  follows 
that  when  we  shall  have  performed  as  many  divisions  as  there 
are  units  less  one  in  the  divisor,  we  will  necessarily  arrive  at  a 
remainder  already  obtained,  (if,  in  fact,  this  remainder  does  not 
recur  sooner).  Now,  annexing  a  0  to  this  remainder,  we  will 
have  a  partial  dividend  exactly  the  same  with  one  of  the  pre- 
ceding ;  whence  it  follows,  that  we  will  have  a  series  of  quotients 
and  remainders  equal  to  the  preceding j  recurring  periodically ^ 
setting  out  from  the  first  partial  dividend,  which  is  equal  to  any 
of  the  preceding. 

Let  us  make  some  applications  of  this. 


GENERAL   PROPERTIES   OF   NUMBERS.  171 

149.  Required  to  reduce  the  fraction,  |,  to  decimals. 
60)7 

'40(0-857142  |  857142 
60 

To 

Yo 
lo 

Here  the  period  shows  itself  after  the  6th  partial  division. 
Second  Example.  —  Let  the  fraction  be  ||. 

130)  37 

~19b(0-331  I  S'SI 

"l3 

In  this  example,  the  period  commences  with  the  fourth  partial 
division. 

Third  Example,  f|. 

290  )  84 


380(0-34523809  |  623809 

lio 


200 
"328 
"680 
"800 


44 

The  period  is  manifest  here  after  the  8th  operation.     But  the 
two  first  decimal  figures  form  no  part  of  the  period,  while  in  the 


172  GENERAL   PROPERTIES   OF    NUMBERS. 

first  two  examples  the  period  commences  with  the  first  decimal 
figure.  The  periodical  decimal  fractions,  whose  period  com- 
mences with  the  first  decimal  figure,  are  called  simple  periodical 
fractions;  and  those  whose  period  commences  after  a  certain 
number  of  decimal  places  already  written,  are  mixed  periodical 
fractions. 

150.  We  have  just  seen  that  certain  vulgar  fractions,  reduced 
to  decimals,  give  rise  to  periodical  decimal  fractions. 

Reciprocally,  every  periodical  decimal  fraction,  simple  or 
mixed,  arises  from,  a  vulgar  fraction,  which  can  easily  he  found 
from  any  given  periodical  fraction. 

This  question  presents  two  distinct  cases ;  either  the  periodical 
fraction  is  simple  or  it  is  mixed.    Let  us  consider  the  first  case. 

Take,  for  example,  the  periodical  fraction 

0-513513513513  

and  let  us  designate  by  N  the  fraction  which  has  given  rise  to  it. 

We  have  N=0-513513513  ....     (1) 

Multiplying  the  two  members  of  this  equation  by  1000,  which  is 
done  in  the  second  member  by  removing  the  point  three  places 
towards  the  right,  we  obtain 

Nx  1000=513-513513513 

Or  (2)  NX  1000=513  +  0-513513513 

Subtracting  (1)  from  (2)  we  have 

NX  999  =  513. 
Then  N=fi|. 

Let  the  fraction  be  N=0-714285714285 (1) 

Multiply  both  members  by  1000000,  we  have 

NX  1000000=714285-714285     (2) 


GENERAL   PROPERTIES   OP   NUMBERS.  173 

Subtracting  the  first  from  the  second, 

Nx  999999=714285 
714285 


N: 


999999 


Reducing  the  fractions  |^|  and  m||f  to  their  simplest 
terms,  we  get 

What  we  have  shown  proves  that  a  simple  periodical  fraction 
is  equivalent  to  a  vulgar  fraction  which  has  for  numerator  the- 
figures  of  the  period,  and  for  denominator  a  number  composed 
of  as  m,any  9'«  as  there  are  figures  in  the  period. 

Thus,  for  an  additional  example,  the  fraction  0'351351351 .... 
is  equivalent  to  the  fraction  |f^=yVT=i|. 

Again,  the  fraction  0-03960396  ....  is  equivalent  to  g|||, 
or  simply  ^W^=TtfT=7^T- 

In  general,  if  a;  =  0,  ahcde  ....  ahcde  ....  ahcde  (where 
abcde  ....  represent  decimal  figures  with  their  relative  values 
and  not  products^,  we  shall  have 

ahcde  .... 
"^^99999 

N.  B.  If  the  periodical  fraction  contains  an  entire  part,  we 
do  not  regard  it  in  forming  the  vulgar  fraction ;  but  we  add  it  to 
the  vulgar  fraction  found  after  it  is  reduced  to  its  simplest  terms. 

Thus,  given  the  periodical  fraction 

4162162 

We  have,  first, 

0-162162  .  .  .  .  =  if|= JA=3^7. 
Then,  4-162162  =  4  +  /^  =  y^*- 

151.   Second  Case.  —  Required  to  find  the  equivalent  vulgar 
fraction,  or  generatrix^  as  it  is  sometimes  called,  of  a  periodical 
mixed  fraction. 
15* 


174  GENERAL   PROPERTIES   OP   NUMBERS. 

Given,  for  example,  the  fraction 

3-45891891  

Multiplying  this  fraction  by  100,  we  obtain  345-891891 ;  and, 
according  to  (N.  B.)  of  preceding  article,  this  expression  has  for 
its  value 

Q..   ,  891         345  X  999+891 
^^^  +  999'^^  999 ' 

345  X  (1000—1)  +  891        345546 
^"■^  999  '  ^^  '~9W~' 

But,  as  we  have  multiplied  the  fraction  by  100,  in  order  to 
reduce  the  result  to  its  true  value,  we  must  divide  it  by  100 ; 
we  thus  obtain  y^^^Yo^,  a  fraction  which,  reduced  to  its  simplest 
form,  becomes  f|f§,  the  generatrix  of  the  mixed  periodical 
fraction,  3-45891891  .... 

If  the  fraction  were  under  the  general  form, 

OfPqrSj  abcde,  abode  .... 

its  value  would  be 

abcde 
^^'■^  + 99999' 

after  multiplying  it  by  10000,  or 

pqrs  X  99999  +  abcde 
99999        ~' 

or,  reducing  the  result  to  its  true  value, 

pqrs  X  99999  +  abcde 
999990000  ' 

We  say,  then,  ani/  mixed  periodical  fraction  whatever  is  equi- 
valent to  a  vulgar  fraction  which  has  for  its  numerator  the 
period,  augmented  by  the  product  of  the  part  which  precedes  the 
period  by  a  number  composed  of  as  many  9s  as  there  are  figures 
in  the  period,  and  for  denominator  this  same  number  of  9.S, 
followed  by  as  many  zeros  as  there  are  figures  in  the  part  which 
precedes  the  period. 


GENERAL  PROPERTIES   OP   NUMBERS.  175 

Take,  for  another  example, 

0-3193069306. 
The  preceding  rule  gives  for  its  value, 

9306  +  31  X  9999  _  9306  +  31  (10000—1)  __ 
999900    .       ~  999900 

309969  +  9306      319275  _  129 
999900        ~  999900  ~  404* 

We  give  here  as  examples  of  simple  and  mixed  periodical 
decimals, 

1st.  0-9999 =1  =  1 

2d.  0-012345679012345679 =  ^V 

3d.  0-987654320987654320 =  ffi 

4th.  16-285714285714 = 

5th.  4-9428571428571  = 

6th.  5-52027027  = 

,  r-o    mi                  •       P7^s  X  99999  +  ahcde  ,     ,    ^ 
152.  The  expression  ^-^ leads  to  some  re- 

markable consequences.     It  can  be  put  under  the  form 

j?grs  (100000  — 1)  +  ahcde 

999990000  ' 

equal  to 

pg-j-sOOOOO — pqrs  +  abcde 
999990000 

This  being  established,  it  is  obvious  from  this  last  form,  that 
if  the  calculations  which  are  indicated  in  the  numerator  are 
effected,  the  result  cannot  be  terminated  by  one  or  more  zeros ; 
for,  in  order  that  this  should  be  the  case,  it  would  be  necessary 
that  some  of  the  last  figures  of  pqrs  should  be  the  same  as  the 
last  figures  of  ahcde;  and,  in  this  case,  the  period  would  not 
commence  after  the  4th  decimal  figure,  as  we  have  supposed. 
(For  example,  if  we  had  s  =  e,  r  =  d,  the  primitive  fraction 
would  be  0,  pqdeabcdeabc  .  .  .  .)  We  s^e,  then,  that  after  the 
reduction  of  the  expression  above  to  its  simplest  terms,  the  result 


176  GENERAL   PROPERTIES   OP   NUMBERS. 

must  be  a  fraction,  wliose  denominator  contains  the  two  factors, 
2  and  5,  or  at  least  one  of  the  two,  to  the  4th  power ;  that  is  to 
say,  to  a  power  whose  degree  is  denoted  by  the  number  of  figures 
which  form  no  part  of  the  period. 

We  can  infer  from  this  the  two  following  propositions : 

1st.  Every  fraction  whose  denominator  does  not  contain  either 
of  the  two  factors  J  2  and  5,  or  is  prime  with  2  and  5,  gives  riscy 
when  reduced  to  decimals,  to  a  simple  periodical  fraction. 

For,  if  we  could  obtain  a  mixed  periodical  fraction,  it  should 
follow,  that  the  equivalent  vulgar  fraction,  which  we  obtain  by 
the  rule  in  (151),  being  reduced  to  its  simplest  terms,  should  be 
equal  to  the  given  fraction.  Now,  that  is  impossible,  (for  in 
order  that  one  irreducible  fraction  be  equal  to  another  fraction, 
the  terms  of  this  last  must  be  the  same  multiples  of  the  terms 
of  the  first).* 

It  results,  then,  that  the  denominator  of  the  proposed  fraction 
would  be  a  multiple  of  2  or  of  6 ;  which  is  contrary  to  the 
hypothesis. 

2d.  Every  irreducible  fraction,  whose  denominator  contains 
one  of  the  factors,  2  and  5,  or  both,  raised  to  a  certain  power, 
gives  rise  to  a  mixed  periodical  fraction,  whose  period  must 
commence  after  we  have  found  as  many  decimal  figures  as  there 
are  units  m  the  greater  of  the  ttvo  exponents  of  2  and  5,  which 
enter  into  the  denominator. 

First,  the  periodical  fraction  cannot  be  simple  ]  for  the  formula 

for  these  sorts  of  fractions  being  „   „„  j  it  is  impossible 

that  this  fraction,  whose  denominator  does  not  contain  either  of 

*  The  terms  of  every  irreducible  fraction  are  prime  with  each  other, 
and  every  fraction  whose  terms  are  prime  with  each  other  is  an  irredu- 
cible fraction.  This  is  obvious,  as  this  reduction  depends  upon  suppress- 
ing the  common  divisor  of  the  two  terms.  Hence,  it  is  obvious,  that  no 
two  irreducible  fractions  can  be  equal,  unless  the  terms  are  identical  in 
both,  nor  can  an  irreducible  fraction  be  equal  to  any  other  fraction  whose 
terms  are  not  the  same  mvltiplen  of  the  terms  of  the  first  fraction. 


GENERAL   PROPERTIES   OF   NUMBERS.  177 

the  factors,  2  and  5,  should  be  equal  to  the  given  fraction  whose 
denominator  contains  these  factors. 

In  the  second  place,  the  period  must  commence  after  n  figures, 
if  n  express  the  greater  of  the  two  exponents  of  2  and  5,  which 
is  found  in  the  denominator;  for  suppose,  for  example,  that  it 
commences  after  n  —  1  figures ;  the  equivalent  to  this  periodical 
fraction  would  have  a  denominator  which  would  only  contain  the 
two  factors,  2  and  5,  or  one  of  them  to  the  (n  —  l)th  power,  and 
could  not  be  equal  to  the  given  fraction,  since  these  two  fractions 
are  supposed  to  be  irreducible. 

For  example,  the  fractions  |,  -l|,  (149),  gave  simple  periodical 
fractions,  because  7  and  37  are  prime  with  2  and  5 ;  but  the 
fraction,  ||,  gave  a  mixed  periodical  fraction,  whose  period  com- 
mences after  the  second  figure,  because  84  is  equal  to  2^X  21. 

Finally,  the  fraction,  j4|^  which  can  be  put  under  the  form 

145 
^^^— pr,  should  give  a  periodical  fraction  whose  period  commences 

after  the  4th  decimal  figure. 

We  find,  in  fact,  for  the  value  of  this  fraction  in  decimals, 
0-8238636636  

153.  We  will  not  carry  farther  the  examination  of  the  proper- 
ties of  periodical  decimal  fractions,  but  close  by  observing  that 
properties  analogous  to  the  preceding  manifest  themselves  in  any 
system  of  numeration  whatever.  The  fractions  in  any  other 
system,  which  enjoy  these  analogous  properties,  are  those  whose 
denominators  are  powers  of  the  Base  of  the«6ystem.  Let  this 
base  be  h. 

First,  in  order  to  reduce  a  vulgar  fraction  into  subdivisions  h 
times  smaller  than  unity,  and  into  other  subdivisions  h  times 
smaller  than  the  first,  &c.,  it  would  be  necessary  to  multiply  the 
numerator  by  6  or  10 ;  that  is  to  say,  to  annex  a  0,  and  divide 
the  result  by  the  denominator ;  which  should  give  in  the  quo- 
tient units  h  times  smaller  than  the  principal  unit,  and  a  certain 
remainder ;  to  write  a  new  0  on  the  right  of  the  remainder,  and 


178  GENERAL   PROPERTIES    OF   NUMBERS. 

divide  the  result  by  the  denominator,  giving  in  the  quotient  units 
h  times  smaller  than  the  preceding,  and  h^  times  smaller  than  the 
principal  unit,  and  so  on.  This  being  established,  we  deduce 
from  it  by  reasoning  precisely  the  same  as  that  which  served  to 
establish  the  properties  of  decimal  fractions  which  arise  from 
vulgar  fractions,  that  the  vulgar  fractions  in  a  system  whose  base 
is  b,  being  converted  into  subdivisions  b,  b^,  c&c,  smaller  than 
unity,  give  rise  to  fractions  (analogous  to  decimals')  of  a  limited 
or  infinite  number  of  figures,  simple  or  mixed  periodical,  and 
that  the  composition  of  the  denominator  of  the  vulgar  fraction 
ivith  reference  to  the  prime  factors  which  enter  into  the  base  b, 
suffices  to  characterize  these  different  sorts  of  fractions. 

We  propose  as  an  exercise  for  the  pupil  the  investigation  of 
the  enunciations  and  demonstrations  of  these  properties. 

Exercises. 

1.  Prove  that  every  entire  even  number  is  the  sum  of  several 
powers  of  2,  and  that  every  entire  odd  number  is  the  sum  of 
several  powers  of  2,  augmented  by  unity. 

Examples,  876,  2539,  6750. 

2.  Every  entire  number,  which  is  not  prime,  has  at  least  one 
prime  divisor  other  than  unity. 

3.  The  remainder  of  the  division  by  9  of  the  product  of  any 
number  of  factors,  is  equal  to  the  remainder  which  the  product 
of  the  remainders  of  the  division  of  each  factor  by  9  gives. 

Prove  that  this  property  belongs  to  every  number,  and  not  to 
9  alone. 

4.  The  product  of  any  three  entire  consecutive  numbers  is 
always  divisible  by  6. 

5.  Convert  the  numbers,  345  and  225,  of  the  decimal  system, 
into  their  equivalents  in  the  binary  system.  Add  these  last  in 
the  binary  system,  and  convert  the  sum  back  to  the  decimal 
system. 


THEORY   OF   RATIOS   AND   PROPORTION.  179 

6.  All  the  prime  numbers,  except  2  and  3,  augmented  or 
diminished  by  unity,  are  divisible  by  6 ;  that  is,  they  are  com- 
prised in  the  general  formula,  Qn  ±  1,  (read  plus  or  minus),  n 
being  any  entire  number. 

7.  If  the  sum  of  the  figures  of  any  number  be  subtracted  from 
the  Bumber  itself,  the  remainder  will  be  divisible  by  9. 

8.  The  expression  rr(n-f  l)(2w+l)  is  always  divisible  by  6. 


CHAPTER  VI. 


APPLICATION  OF  THE  RULES   OF  ARITHMETIC. - 
THEORY  OF  RATIOS  AND  PROPORTION. 

154.  Introduction.  —  We  have  seen,  in  the  course  of  the 
explanation  of  the  different  operations  of  arithmetic,  that  these 
operations  give  rise  to  two  principal  species  of  questions.  1st. 
Those  which  have  for  their  object  to  demonstrate  the  existence 
of  certain  properties  of  certain  numbers  known  and  given.  2d. 
Those  in  which  it  is  proposed  to  find  certain  numbers  from  the 
knowledge  of  other  numbers  having  fixed  relations  with  the  first. 
The  first  are  theorems,  properly  speaking ;  but  we  have  generally 
called  them  Principles  and  Propositions.  The  questions  of  the 
second  species,  which  are  not  particular  applications  of  the  rules 
and  principles,  are  called  Problems. 

The  Problems,  which  we  have  hitherto  solved,  have  been  easy 
of  solution,  because  the  data  were  simple,  and  the  relations  be- 
tween the  known  and  unknown  quantities  very  obvious.  But 
this  is  not  generally  the  case  -,  as  very  often,  in  order  to  arrive  at 
a  solution,  we  have  a  considerable  difficulty  to  overcome,  which 
consists  in  discovering  and  determining  the  series  of  operations 
to  be  executed  upon  the  numbers  known  and  given,  in  order  to 
arrive  at  a  knowledge  of  the  numbers  sought. 


180  THEORY   OF  RATIOS   AND   PROPORTION. 

Nevertheless,  there  exists  a  certain  class  of  questions,  the  re- 
solution of  which  can  be  subjected  to  fixed  and  certain  rules ; 
these  are  particularly  those  in  which  we  consider  Proportional 
Magnitudes. 

The  greater  part  of  these  questions  are  precisely  those  which 
the  general  necessities  of  society  give  rise  to,  in  that  which  relates 
to  its  commercial,  industrial,  and  financial  interests;  they  are 
generally  known  as  the  Rule  of  Three^  the  Rules  for  the  calcula- 
tion of  Interest,  Discount,  the  Rule  of  Fellowship,  Excharige,  &c. 

To  arrive  easily  at  the  solution  of  these  questions,  we  will 
commence  by  explaining  the  theory  of  ratios  and  proportions. 

§  I.  —  Of  Ratios  and  Proportions,  and  of  their  Prin- 
cipal Properties. 

155.  We  have  already  said  (1),  that  in  order  to  form  an  idea 
of  any  magnitude  whatever,  we  must  compare  it  with  some  other 
magnitude  agreed  upon,  of  the  same  species,  which  can  be  taken 
arbitrarily  or  in  nature.  The  result  of  this  comparison  is  what 
we  have  called  number.  Number,  then,  expresses  the  relation 
between  any  magnitude  and  its  unit.  Now,  if  we  wish  to  com- 
pare any  two  magnitudes  whatever,  of  the  same  species,  or  what 
is  the  same  thing,  to  compare  the  numbers  which  express  them, 
the  result  of  this  comparison  is  a  relation  between  these  two 
numbers. 

When  we  thus  compare  two  magnitudes  with  each  other,  we 
may  either  wish  to  know  hoio  much  the  greater  exceeds  the  less, 
or  how  many  times  the  greater  contains  the  less.  From  this 
results  two  sorts  of  relations  between  the  numbers  compared, 
one  which  is  sometimes  called  an  Arithmetical  ratio,  and  another 
called  a  Geometrical  ratio.  But  these  names,  which  are  but 
little  significant,  are  well  replaced  by  the  word  difference,  in 
order  to  express  the  result  of  the  comparison  by  subtraction,  and 
Ratio  to  express  the  result  of  the  comparison  by  division. 

Thus,  let  24  and  6  be  the  two  numbers  which  we  wish  to 
compare.  We  have  24  —  6  =  18  for  the  difference,  and  ^^  =  4 
for  the  Ratio. 


THEORY   OF  RATIOS   AND   PROPORTION.  181 

The  relations  of  magnitudes  by  division  or  Ratios  will  chiefly 
occupy  the  present  chapter,  as  by  far  the  most  important  of  the 
two  classes  of  relations ;  but  we  will  first  give  one  or  two  leading 
properties  of  the  Relations  hi/  Subtraction  or  Differences. 

156.  In  every  Difference  or  Ratio,  the  two  terms  are  thus  dis- 
tinguished. The  one  first  written  is  the  antecedent ;  the  second 
term  is  the  consequent.  Thus,  in  the  expressions  24  —  6,  \^, 
24  is  the  antecedent  in  both  cases,  and  6  is  the  consequent.  When 
the  difference  between  two  numbers  is  equal  to  the  difference  be- 
tween two  other  numbers,  the  four  numbers  taken  together  form 
an  Equi-difference, 

For  example,  let  the  four  numbers,  12,  5,  24,  17 ;  the  differ- 
ence of  12  and  5  is  7 ;  the  difference  of  24  and  17  is  also  7. 
These,  then,  form  an  equi-difference  which  We  write  thus : 

12.5  :  24.17. 

Placing  one  point  between  1st  and  2d  terms,  two  points  between 
2d  and  3d,  and  one  between  the  3d  and  4th.     We  enunciate  it 

12  is  to  5  as  24  is  to  17; 

that  is,  12  exceeds  5  by  as  many  units  as  24  exceeds  17.  We 
can  also  write  it 

12  —  5  =  24  —  17; 

12  and  24  are  the  antecedents  ;  5  and  17  the  consequents.  The 
first  and  last  term  are  moreover  called  the  extremes  ;  the  second 
and  third  the  means. 

This  established,  we  say  that,  in  every  equi-difference^  the  sum 
of  the  extremes  equals  the  sum  of  the  means. 

Let  11.7:  19.15; 
We  have  obviously  11  +  15  =  7  -f  19. 

To  prove  this  generally,  we  observe  that  if  the  consequents 
were  equal  to  their  antecedents,  as  for  example, 

11.11  :  19.19, 
16 


182        THEORY  OF  RATIOS  AND  PROPORTION. 

the  proposition  would  be  manifestly  true.  Now,  in  order  to  place 
the  first  equi-difference  under  this  form,  we  have  simply  to  add  4 
to  each  of  its  consequents ;  that  is,  the  sum  of  the  means  and 
sum  of  the  extremes  are  augmented  by  the  same  number.  Hence, 
if  these  sums  are  equal  now,  they  must  have  been  so  before. 
Then,  &c. 

As  a  consequence  of  this  property,  knowing  three  terms  of  an 
equi-difference,  we  can  find  the  fourth.     Thus,  let 

23.11  :  49.x,  (x  being  the  unknown  term), 

be  the  equi-difference,  we  have 

23  -i-  a;  =  49  -f  11 ; 

whence  x  is  known.  Sometimes  two  of  the  terms  of  the  equi- 
difference  are  the  same  as 

27.39  :  39.51. 

Here  the  double  of  one  of  the  means  is  equal  to  the  sum  of 
the  two  extremes,  or  the  mean  itself  is  equal  to  half  the  sum  of 
the  extremes.     Thus,  in  the  equi-difference, 

23..X  :  a:.49, 
^^49+23^g^^ 

and  this  number  is  called  the  average  or  arithmetical  mean  of 
the  two  numbers. 

It  is  useless  to  proceed  farther  with  the  properties  of  equi- 
differences,  as  they  are  of  very  little  use.  We  will  merely  add, 
that  no  transformation  executed  upon  an  equi-difference  destroys 
this  equi-difference,  so  long  as  the  sum  of  the  extremes  remains 
equal  to  the  sum  of  the  means. 

We  pass  to  the  discussion  of  Ratios  and  Proportions,  properly 
so  called. 

157.  The  ratio  of  two  magnitudes,  we  have  seen,  is  the  quo- 
tient of  the  division  of  the  numbers  which  express  these  magni- 
tudes.    This   ratio   can   be   an  entire  number  or   a  fractional 


THEORY   OF  RATIOS   AND   PROPORTION.  183 

number,  greater  or  less  than  unity.    For  example,  the  ratio  of  24 
to  6  is  \S  or  4 ;  that  of  6  to  24  is  g^,  or  | ;  that  of  75  to  18  is 

7  5     nr  2  5 

It  is  in  the  sense  of  Ratio  that  we  have  hitherto  understood 
the  comparison  of  any  magnitude  whatever  with  its  unit  (No.  1). 
In  the  theory  of  compound  numbers,  the  relation  of  the  princi- 
pal unit  to  its  subdivisions,  or  between  two  subdivisions,  is  the 
number  of  times  which  the  one  contains  the  other. 

158.  The  comparison  of  two  concrete  numbers  supposes  always 
that  these  magnitudes  are  of  the  same  species,  since  we  cannot 
compare  magnitudes  of  diflferent  species  (No.  2). 

The  ratio  is  itself,  by  its  very  definition,  essentially  an  abstract 
number,  expressing  how  many  times  one  of  the  numbers  contains 
the  other,  or  is  contained  in  it.  The  antecedent  and  consequent, 
which  form  the  latio,  are,  we  have  seen,  the  numerator  and  de- 
nominator of  a  fractional  expression  which  we  obtain,  in  indi- 
cating the  division  of  the  two  magnitudes  which  we  are  com- 
paring. 

159.  When  the  ratio  of  two  numbers  is  equal  to  the  ratio  of 
two  other  numbers,  we  say,  that  the  four  numbers  or  magnitudes 
which  they  represent,  are  in  proportion,  or  proportional.  A 
proportion  is  then  the  expression  of  the  equality  of  two  ratios. 

For  example,  the  ratio  of  48  to  12  being  4,  and  of  86  to  9 
being  also  4,  we  have  the  equation 

48  =  3^6^  or  48  :  12  =  36  :  9. 

It  is  sometimes  more  convenient  to  present  the  proportion 
under  the  form  48  :  12  :  :  36  :  9,  which  is  thus  enunciated : 

48  is  to  12  as  36  is  to  9. 

The  terms  48  and  36  are  antecedents ;  12  and  9  are  conse- 
quents. The  first  and  fourth  are  extremes  ;  the  second  and  third 
are  means. 


184        THEORY  or  RATIOS  AND  PROPORTION. 

160.  Fundamental  Properti/.  —  All  proportions  possess  a  pro- 
perty which  may  serve  as  a  basis  for  the  resolution  of  the  pro- 
blems whose  enunciations  contain  j^roportional  quantities.  This 
property  consists  in  this  : 

In  every  proportion  the  'product  of  the  extremes  is  equal  to  that 
of  the  means. 

Let  the  proportion  be 

(1)     24  :  18  T:  20  :  15, 

of  which  the  ratios  f  |  and  f  §  each  equals  |.  We  say,  that  we 
must  have 

24  X  15  =  18  X  20. 

For  the  property  would  be  evident  if  we  had  the  proportion 
24  :  24  :  :  20  :  20,     (2) 

(which  we  call  an  identical  proportion).  Now,  to  render  the 
proportion  (1)  the  same  as  (2),  it  suffices  obviously  to  multiply 
each  consequent  by  | ;  but  by  this,  we  multiply  the  product  of 
the  extremes  and  the  product  of  the  means  by  the  same  number, 
and  make  the  same  change  in  both.  Hence,  if  equal  after  the 
multiplication,  they  must  have  been  equal  at  first.  Hence  the 
property  is  proved. 

161.  Reciprocally.  — If  the  product  of  two  numbers  is  equal 
to  the  product  of  two  other  numbers,  these  four  numbers  form  a 
proportion  of  which  either  pair  of  factors  will  constitute  the 
means,  the  other  pair  constituting  the  extremes. 

For,  if  no  proportion  existed  among  these  four  numbers,  it 
would  be  necessary,  in  order  to  render  the  second  and  fourth 
respectively  equal  to  the  first  and  third,  to  multiply  each  one  by 
a  different  number,  expressing  in  the  one  case  the  ratio  of  the 
first  term  to  the  second,  in  the  other  of  the  third  to  the  fourth ; 
and  as  the  two  products  would  thus  become  equal  by  the  multi- 
plication of  each  by  a  different  mimber,  it  would  result  that  they 
were  not  equal  before  the  multiplication ;  which  would  be  con- 
trary to  the  enunciation  of  the  proposition. 

Then,  &c.,  &c. 


THEORY   OF  RATIOS    AND   PROPORTION.  185 

162.  Another  demonstration  of  the  fundamental  property  and 
its  reciprocal.  (We  employ  letters  in  order  to  render  the  reason- 
ing more  concise  and  general). 

Let  a,  h,  c,  d,  be  four  numbers  in  proportion,  so  as  to  give 

a  :  b  '.  :  c  :  a,  or  -j-=  -j. 
o        a 

If  we  multiply  the  two  members  of  this  equality  hj  h  x  d, 
product  of  the  two  consequents,  we  obtain 
a  X  b  xd  _c  X  b  xd 
b  ^         d        • 

Suppressing  in  each  member  the  factor  common  to  the  numera- 
tor and  denominator,  we  have 

a  X  d  —  c  X  b. 
Then  the  product  of  the  extremes  is  equal  to  that  of  the  means. 
Reciprocally,  let  the  four  numbers,  a,  b,  c,  c?,  be  such,  that 
we  have 

a  xd  =  b  X  c. 

Let  us  divide  the  two  members  of  this  equality  hy  b  X  d, 
product  of  one  factor  of  the  first  member  by  one  factor  of  the 
second,  we  have  thus 

a  X  d  _b  X  c 

b  X  d~  bx  d^ 
or,  simplifying, 

-^  =  -7,  or  a  :  6  :  :  c  :  a. 

0        a 

Thus,  the  /our  numbers  form  a  proportion  of  which  the  factors 
of  the  first  product  constitute  the  extremes,  the  factors  of  the  second 
product  the  means. 

163.  First  Consequence.  —  In  every  proportion  we  can  cause, 
1st,  the  tivo  means  to  exchange  places;  2d,  the  two  extremes  to 
change  places  J  3d,  the  means  to  exchange  places  with  the  extremes 
witJiout  destroying  the  proportion  between  the  four  numbers  thus 
written. 

16* 


186  THEORY   OF   RATIOS   AND   PROPORTION. 

For  it  is  evident  that  these  changes  do  not  alier  the  equality 
of  the  two  products  which  the  extremes  and  means  of  the  primi- 
tive proportion  give.  And  since,  in  the  new  expressions,  the 
product  of  the  first  number  by  the  last  always  remains  equal  to 
the  product  of  the  second  by  the  third.,  there  will  always  exist  a 
proportion  between  the  four  numbers  after  the  changes  are 
effected. 

Let  the  proportion  be,  for  example, 

48  :  36  :  :  72  :  54.  (1) 
We  have,  by  changing  the  means  for  each  other, 

48  :  72  :  :  36  :  54.  (2) 
By  exchanging  the  extremes, 

54  :  36  :  :  72  :  48.     (3) 

By  placing  extremes  in  the  places  of  the  means,  and  the  means 
in  the  places  of  the  extremes, 

36  :  48  :  :  54  :  72.     (4) 

In  the  expressions  (2),  (3),  (4),  the  product  of  the  second 
number  by  the  third,  is 

36  X  72,  or  48  X  54; 

and  the  product  of  the  first  by  the  fourth, 

48  X  54,  or  36  X  72. 

Now,  these  products  are  equal  by  virtue  of  proportion  (1) ;  then 
the  expressions  (2),  (3),  and  (4),  are  also  proportions. 

The  common  ratio  of  (1)  is  |,  of  (2),  |,  of  (3),  |,  and  |  for 
the  proportion  (4). 

N.  B.  It  is  obvious  that  inverting  the  order  of  the  terms  in 
each  ratio  does  not  destroy  the  proportion,  since  it  amounts  to 
the  same  change  as  is  exhibited  in  (4). 

164.  Second  Consequence.  —  We  can  in  every  froportion 
multiply  or  divide  one  extreme  and  one  mean  by  the  same  num- 
ber, without  destroying  the  proportion. 


THEORY   OF   RATIOS   AND   PROPORTION.  187 

For  the  products  of  the  extremes  and  means  of  the  given 
proportion  being  equal,  the  new  products  which  result  from  the 
multiplication  or  division  of  these  products  by  the  same  number 
will  also  be  equal;  and  the  proportion  will  still  exist.  There 
are  many  other  properties  of  proportions ;  but  those  which  we 
have  just  developed  are  the  only  ones  of  which  we  shall  have 
need  for  the  resolution  of  the  problems  which  depend  on  this 
theory. 

§  II.  —  Resolution   op   Questions   dependent  on   the 
Theory  of  Proportional  Quantities. 

Rule  of  Three. 

165.  A  great  number  of  problems  in  commerce,  banking,  &c., 
contain  in  their  enunciation  numbers  bearing  relations  to  each 
other  susceptible  of  being  expressed  by  proportions.  Of  these 
numbers  some  are  given  and  known,  the  others  unknown,  to  be 
determined.  We  designate,  under  the  title,  the  Rule  of  Three, 
the  process  by  which  we  find  the  fourth  term  of  a  proportion 
when  three  terms  are  given. 

Now,  from  the  property  of  every  proportion  that  the  product 
of  the  extremes  is  equal  to  the  product  of  the  means,  it  results 
necessarily  that,  in  order  to  obtain  the  value  of  the  unknown 
term,  we  must,  if  it  is  an  extreme,  divide  the  product  of  the 
means  hy  the  known  extreme. 

And  if  it  is  a  mean,  we  must  divide  the  product  of  the  ex- 
tremes hy  the  known  mean. 

Thus,  let  the  two  proportions  be 

24  :9  :  :32  :cc;  45  :36  :  ::«  :24; 
(we  denote  the  unknowns  by  the  last  letters  of  the  alphabet). 
Since  the  first  gives  24  X  cc  =  9  X  32,  there  results 

9x32       ^^ 
^=--24 ^2' 


188  THEORY   OF   RATIOS   AND   PROPORTION. 

we  have  also  for  the  second, 

SQx  X  =  4:5x24:. 

Whence,  :r  =  l^^  =  15|?  =  30. 

The  proportions  become  then 

24  :9  :  :  32  :  12;  45  :  36  :  :  30  :  24. 

The  common  ratio  is  |  for  the  first,  and  |  for  the  second. 

We  pass  now  to  the  resolution  of  some  problems,  of  which 
those  in  (41)  may  be  considered  particular  examples. 

166.  Problem  First.  —  Required,  the  price  of  384  lbs.  of  a 
certain  commodity ,  2b  lbs.  of  which  cost  ^650? 

Analysis.  —  Since  25  lbs.  cost  $6*50,  it  is  clear  that  2,  3,  4 

times  25  lbs.  must  cost  2,  3,  4  ...  .  times  as  much ;  thus,  the 
two  given  numbers  of  pounds  bear  to  each  other  the  same  rela- 
tion as  their  respective  prices.  Then,  if  we  designate  by  x  the 
unknown  price  of  384  lbs.,  and  if  we  consider  for  the  moment 
the  three  given  numbers  and  x  as  abstract  numbers,  we  have 
the  proportion 

(1)    25  :  384  :  :  650  :  x. 

Whence  (165),  .  ==  ''%'''  =  ^  =  9984 ; 

and  we  conclude  that  the  384  lbs.  of  the  commodity  ought  to 
cost  $99-84. 

N.  B.  Before  seeking  the  value  of  x  by  means  of  the  propor- 
tion (1),  we  can  simplify  that  proportion  in  observing  that  the 
antecedents,  that  is,  one  extreme  and  one  mean,  are  divisible  by 
25.     We  then  suppress  this  factor  (164),  and  obtain 

1  :  384  :  :  26  :  a: ;  whence,  cc  =  384  X  26  =  9984. 

Another  method  of  resolution.  —  If  2b  lbs.  cost  $6-50,  one 

pound  must  cost  25  times  less,  or  -^^  of  $6-50;  that  is,  — ijr-- 


THEORY   OF   RATIOS   AND   PROPORTION.  189 

Then,  SS4:lbs.  will  cost  384  times  as  much  as  1  lb.,  or  X 

384;  which  gives  899-84. 

Second  Prohlem.  —  It  takes  135  men  20  days  to  do  a  certain 
piece  of  work  ;  how  many  days  would  300  men  require  to  per^ 
form  the  same  labour  ? 

Analysis.  —  If  a  certain  number  of  men  have  employed  20 
days  in  accomplishing  the  work,  it  is  clear  that  a  number  of  men 
2,  3,  4  ...  .  times  as  great  must  occupy  2,  3,  4  ...  .  times 
shorter  period  to  do  the  same  work,  other  things  being  equal ; 
then,  as  many  times  as  the  first  number  of  men,  135,  is  contained 
in  the  second  number,  300,  so  many  times  the  number  of  days 
necessary  for  the  second  number  of  men,  or  the  number  sought, 
Xj  will  be  contained  in  the  number  of  days  necessary  for  the  first 
number  of  men. 

Thus,  we  have  the  proportion 

135  :300:  :  a;  :  20; 

whence,  (165),  x  =  — 390"  ^  ^' 

Then,  it  takes  300  men  9  days  to  do  the  work.  We  could  have 
suppressed  in  this  proportion  the  factor,  15,  common  to  the  two 
first  terms,  and  the  factor,  20,  common  to  the  two  consequents. 

We  should  then  have 

1  :  9  :  :  1  :  a:;  whence,  x  =  9. 

Another  mode  of  resolution.  — If  135  men  took  20  days  to  do 
the  work,  it  would  have  taken  one  man  135  times  as  much  time, 
or  135  X  20  days,  and  300  men  would  have  required  a  number 
of  days  300  times  as  small  as  20  x  135 ;  that  is  to  say, 

20x135       2700      _, 


190        THEORY  OF  RATIOS  AND  PROPORTION. 

Ratios,  Direct  and  Inverse. 

1G7.  Before  treating  more  complicated  problems,  we  must 
make  known  certain  terms  which  the  consideration  of  propor- 
tional quantities  give  rise  to. 

In  every  question,  the  enunciation  of  which  contains  four 
numbers  in  proportion,  two  of  these  numbers  are  of  a  certain 
species,  and  the  two  others  of  another  species ;  but  each  term 
of  the  second  species  is  closely  connected  by  the  conditions  of 
the  question  with  one  of  the  terms  of  the  first. 

It  is  thus  in  the  first  problem  (166),  two  of  the  four  numbers 
express  the  weights  of  a  certain  commodity ;  the  other  two,  the 
respective  prices  of  these  weights. 

In  the  same  manner,  in  the  second  problena,  we  had  two  num- 
bers of  men,  and  two  numbers  of  days;  and  the  latter  expressed 
the  respective  periods  employed  by  the  two  numbers  of  men  to 
do  the  same  work.  It  is  agreed,  for  this  reason,  to  call  the  two 
terms  of  difi"erent  species,  thus  connected  by  the  enunciation  of 
the  question,  Correspondents. 

For  example,  in  the  first  problem,  the  prices  are  the  corre- 
spondents of  the  pounds ;  and  vice  versa,  the  numbers  of  pounds 
are  the  correspondents  of  the  prices. 

This  established,  we  say,  that  there  is  a  Direct  Relation  be- 
tween the  numbers  of  the  first  species  and  the  numbers  of  the 
second ;  or  that  these  numbers  are  directly  proportional,  when, 
the  proportion  having  been  established,  we  see  that  as  each 
number  increases  or  diminishes,  its  correspondent  increases  or 
diminishes;  and  that,  on  the  contrary,  the  Relation  is  Inverse, 
or  the  four  numbers  are  inversely  (or  reciprocally)  proportional, 
when,  as  each  number  increases  or  diminishes,  its  correspondent 
diminishes  or  increases. 

The  enunciation  of  the  first  problem  oifcrs  the  example  of  a 
direct  relation;  for  the  greater  the  number  of  pounds,  the 
greater  the  price. 


THEORY   OP   RATIOS   AND   PROPORTION.  191 

The  second  problem  gives  rise  to  an  inverse  relation  ;  for  the 
more  men  there  are  to  do  the  work,  the  shorter  period  required. 

If  the  relation  is  direct^  and  if  we  wish  to  write  the  propor- 
tion under  the  form 

a  '.  h  '.  \  c  ',  dj 

one  of  the  numbers,  and  its  correspondent^  must  form  the  two 
antecedents,  and  the  other  two  the  consequents.  On  the  contrary, 
if  the  relation  is  inverse,  one  of  the  numbers  and  its  correspondent 
must  form  the  extremes,  while  the  other  two  form  the  two  means. 
When  we  write  the  proportion  under  its  equivalent  form  of  two 
equal  fractions, 

a        c 
T'^d' 

it  is  necessary,  in  the  case  of  the  direct  relation,  that  one  of  the 
numbers  and  its  correspondent  form  the  two  terms  of  the  first 
fraction,  or  the  numerators  of  the  two  fractions,  while  the  other 
numbers  form  the  two  terms  of  the  second  fraction,  or  the  deno- 
minators of  the  two  fractions ;  and,  in  the  case  of  the  inverse 
relation,  each  number  and  its  correspondent  must  form  the 
numerator  of  the  first  fraction  and  the  denominator  of  the 
second,  or  the  denominator  of  the  first  fraction  and  the  numera- 
tor of  the  second. 

N.  B.  All  these  distinctions  in  the  manner  o^.  writing  the 
proportions  furnished  by  the  enunciations  of  the  problems  are 
of  importance,  and  should  be  carefully  retained  in  the  memory. 

168.  We  say,  also,  when  the  relation  is  direct,  that  one  quan- 
tity of  each  species  is  in  direct  proportion  with  its  correspondent ; 
and  if  the  relation  is  inverse,  that  each  quantity  is  in  inverse 
proportion  with  its  correspondent.     Thus,  for  example. 

Two  fractions  of  the  same  denominator  are  in  direct  propor- 
tion with  their  numerators. 

For  we  have  seen  that  if  the  numerator  is  rendered  double, 
triple,  quadruple,  ....  or  one  half,  one  quarter,  one  third  of 


192        THEORY  OP  RATIOS  AND  PROPORTION. 

what  it  is,  the  fraction  will  be  rendered  two,  three,  four  .... 
times  greater  or  less  than  it  was. 

By  an  analogous  process,  we  could  prove,  that  two  fractions, 
having  the  same  numerators,  are  in  the  inverse  proportion  of 
their  denominators. 

When  the  fractions  have  different  numerators  and  denomina- 
toi'S,  we  commence  by  reducing  them  to  the  same  denominator 
or  to  the  same  numerator,  and  the  question  is  thus  reduced  to 
one  of  the  two  preceding  cases. 

We  are  then  led  to  a  new  mode  of  expression,  which  consists 
in  saying  that  the  given  fractions  are  in  Compound  Proportion, 
direct  or  inverse,  of  the  two  products  of  the  numerator  of  the 
Jlrst  by  the  denominator  of  the  second,  and  of  the  numerator 
of  the  second  by  the  denominator  of  the  first. 

In  order  to  justify  this  mode  of  expression,  let  us  consider, 
for  example,  the  two  fractions,  |  and  -^^. 

Reducing  them  to  the  same  denominator,  we  obtain 

3  X  11       J    4  X  7 
and 


7  X  11  7  X  11' 

and  these  two  fractions  are  in  the  direct  ratio  of  3  X  11  to  4  X  7, 
or  of  33  to  28.  If,  on  the  contrary,  we  reduce  them  to  the  same 
numerator,  they  bcQome 

3x4       ,3x4 
and 


7x4  3  x  11' 

and  in  this  case  the  two  fractions  are  in  the  inverse  ratio  of  the 
denominators,  or  the  first  is  to  the  second  as  3  x  11  is  to  7  X  4, 
or  as  33  is  to  28,  the  same  as  before. 

But  we  see  that  the  two  terms  of  this  ratio  are  the  one,  the 
product  of  the  numerator  of  the  first  fraction  by  the  denomina- 
tor of  the  second ;  the  other,  the  product  of  the  numerator  of 
the  second  by  the  denominator  of  the  first. 

This  compound  ratio  is  in  some  sort  the  result  of  the  multipli- 
cation of  two  simple  ratios,  which  are  either  direct  or  inverse 
with  regard  to  each  other. 


THEORY   OF   RATIOS   AND   PROPORTION.  193 

169.  Ap2)licat{ons.  —  As  application  of  what  has  just  been 
said,  we  will  indicate  the  method  of  bringing  into  a  proportion 
certain  surfaces  and  volumes  or  solids,  because  there  is  a  number 
of  questions  in  which  we  have  need  of  these  numerical  valua- 
tions. 

Let  it  be  required  to  compare  the  superficial  extent  of  two 
pieces  of  stuff,  one  of  which  is  24  yards  long  by  |  yard  wide  ; 
the  other,  17  yards  long  by  |  wide. 

By  a  process  of  reasoning  analogous  to  that  used  in  (166),  we 
see  that  24  yards  long  by  |  yards  wide.j  is  the  same  thing  as 
24  X  I  yards  loiig  by  1  yard  wide. 

In  the  same  manner,  17  yards  long  by  j  of  a  yard  wide,  is 
equal  to  17  X  |  long  by  1  yard  wide. 

Then,  since  the  breadth  of  the  two  pieces  is  the  same,  the 
ratio  of  the  two  superficial  extents  is  equal  to  that  of  the  two 
lengths,  and  we  lind  this  ratio  to  be 

94x^-17x=   „,24x2.17x5       24x2x4.3x17x5. 
-4X,.17x„or      3     .     ^     '"'        3xT-     3x4    ' 

or,  simplifying,  as  64  :  85. 

Again,  let  there  be  two  rolls  of  paper-hangings,  one  of  which 
is  15  yards  long,  by  |  of  a  yard  wide ;  the  other  19  yards  long, 
by  I  of  a  yard  wide ;  we  would  find  in  the  same  manner  for  the 
ratio  of  the  superficial  extents  of  the  two  rolls, 

-.r    /    in     n       15x4  19x7        15x4x8   19x7x5 
15xf  :19xi,  or  -^:-g-,  or  -^_^:_^-^;  or, 

simplifying,  96  :  133. 

We  conclude  from  this,  that,  whenever  the  enunciation  of  a 
question  gives  rise  to  a  comparison  of  superficial  extents,  in  order 
to  reduce  them  to  the  unit  of  length,  we  must  form  the  product  of 
the  length  hy  the  breadth,  and  then  compare  the  resultitig  quan- 
tities. 

As  to  volumes  or  solids,  it  will  suffice  to  take  one  example,  in 
order  to  determine  the  steps  to  be  followed. 
17 


194  THEORY   OF   RATIOS   AND   PROPORTION. 

Required  to  determine  the  ratio  in  cubic  yards  of  the  solid 
contents  of  two  pieces  of  masonry  ? 

We  suppose  that  the  first  piece  is  60  yards  long,  by  |  of  a 
yard  thick,  and  3  yards  high;  and  the  second,  125  yards  long, 
by  I  of  a  yard  thick,  and  4^  yards  high. 

Reasoning,  as  in  the  preceding  case,  we  find  that  for  the  first 
wall  it  is  as  if  it  was  60  x  |  X  3  yards  long,  by  1  yard  thick,  and 
1  yard  high ;  and  for  the  second,  125  X  |  X  |  yards  long,  by  1 
yard  thick,  and  1  yard  high.  In  other  words,  the  two  walls  must 
contain,  respectively,  60  X  |  X  3  cubic  yards,  and  125  X  |  X  | 
cubic  yards.  Then,  the  ratio  of  the  two  volumes  is  equal  to  that 
of  these  two  products,  or  of 

60x3x3x4^    125x7x9  .  .^  ,    ,^. 

T7^ to  :r5 ,  or  of  48  to  175. 

lb  lb 

Whence  we  see,  that  in  order  to  obtain  the  two  pieces  of  work 
expressed  in  cubic  yards,  it  suffices  to  form  for  each  one  of  them 
the  product  of  the  length  by  the  thickness  and  by  the  height. 
After  which  we  .easily  find  the  ratio  of  the  two. 

Compound  Rule  of  Three.  —  General  Method  of  Reduction  to 

Unity. 

170.  The  enunciation  of  a  question  often  contains  more  than 
four  numbers,  between  which  it  becomes  necessary  to  establish 
either  direct  or  inverse  proportions ;  and  thus  arise  the  distinc- 
tions, Single  Rule  of  Three,  and  Compound  or  Double  Rule  of 
Three.  These  names  arise  from  the  mode  of  resolution,  which 
is  an  application  of  the  theory  of  proportions.  But  this  mode 
has  been  generally  replaced  by  the  method  called  the  method  of 
Reduction  to  Unity,  which  we  will  now  develop,  remarking  that 
the  second  mode  of  resolution  of  the  problems  iu  (166)  is  a  par- 
ticular case  of  this  general  method. 

171.  Third  Problem.  —  It  requires  1800  yards  of  cloth,  |  of 
a  yard  wide,  to  clothe  500  men.  Required  the  number  of  yards 
of  cloth,  \of  a  ya.rd  wide,  which  shall  clothe  960  men  ? 


THEORY   OF  RATIOS   AND   PROPORTION.  195 


Table  of  Calculation. 


1800  yards  long,    |  wide,  500  men. 


X 


I     "      960 


1800  x|       ''         1     "      500     " 
XX I  "         1     '^      960     " 


1800x5 
4x500 
xxl 
960x8 


la  la 

i      "  1      " 


Then      ^X7    _1800x5 
■"       ^  960  X  8       4  X  500  * 

Analysis.  —  After  arranging  upon  two  horizontal  lines,  the  six 
numbers  which  the  enunciation  contains,  and  of  which  the  num- 
ber of  yards  required  forms  part,  we  reason  in  the  following 
manner :  1800  yards  long^  by  |  wide,  and  x  yards  long,  by  | 

1800x5        .xxl       ^    , 
"Wide,  are  the  same  thing  as  ^ ,  and  — ^  yards  long,  by 

1  yard  wide. 

We  write,  then,  these  numbers  upon  two  new  lines,  preserving 
the  numbers  960  and  500  in  their  respective  places  in  the  two 

new  lines.     Since,  with  ^ yards  long,  by  1  yard  wide, 

we  can  clotbe  500  men,  one  man  could  be  clothed  with  —. — ^777^- 

4  X  500 

X  X  1 
In  the  same  manner,  if  — ^    yards  can  clothe  960  men,  one 

X  X  7 
man  could  be  clothed  with  r: — ^77777,  which  eives  a^ain  two  new 
8x960  ®  ^ 

lines,  which  we  place  below  the  preceding.     Now,  the  two  last 

expressions  which  we  have  just  obtained,  representing  both  the 

quantit}'-  of  cloth  necessary  to  clothe  one  man,  are  necessarily 

equal.     We  have  then 

XX  7    _  1800  X  5 
8  X  960  ~~  4x500  ' 


196  THEORY   OP   RATIOS   AND   PROPORTION. 

or,  reducing  to  the  same  denominator,  and  then  suppressing  this 
denominator, 

X  X  7  X  4  X  600  =  1800  X  5  X  8  X  960. 

Dividing  the  two  members  of  the  equation  by  the  multiplier 
of  Xj  we  have 

_1800x5x8x960_36x960_  34560 _ 
^"       7x4x500        -        T~ 7--49^7^; 

that  is,  it  would  require  4937^  yards  to  clothe  960  men. 

Verification. 

1800x5      ,  v  .      1     .    ifi        .1 

4^5QQ  reduces,  obviously,  to  Ls^  or  4^; 


on  the  other  hand, 

34560 


X 


7         8  X  960' 

reduces  also  to  4^.  The  number  4^,  or  4  yards  and  a  half,  ex- 
presses in  the  two  cases  the  quantity  of  cloth  necessary  to  clothe 
one  man. 

172.  Problem  Fourth.  —  500  men,  icorldng  12  Jiours  a  day, 
employed  57  days  in  excavating  a  canal  1800  yards  long,  hy  7 
yards  wide,  hy  3  yards  deep  ;  required  in  how  many  days  860 
men,  working  10  hours  a  day,  can  dig  another  canal  2900  yards 
long,  hy  12  wide,  and  5  deep,  in  an  earth  3  times  as  difficult  to 
excavate  as  the  first.  (This  is  one  of  the  most  complicutcd  ques- 
tions which  can  be  given  in  this  Compound  Proportion,  or  Rule 

of  Three.) 

Tahle  of  Calculations. 

500  men.     12  hours.     57  days.    (1800x  7x3x  1)  cubic  yards. 

860     "       10     "  X      "     (2900x12x5x3)  '' 

_  ,,        51800x7x3x11 

Iman        1  hour         1  day     \   500^1^^57} 

1     u  1     .  ^     u      ^2900X12X5X3|         ,, 

^  ^  ^  \        860x10        i 


THEORY   OF   RATIOS   AND   PROPORTION.  197 

2900x12x5x3   ^.  .,  ^^    1800x7x3x1     ,,^ 
^^^^°^  ^= 860x10        -  ^'^'^'^  ^y    500x12x57    '  ^^^ 

_  2900x12x5x3x500x12x57 
or,  X  —      ^^,^ ^^,^^  ^g^^  ^-^  ^g  ^      . 

Analysis.  —  It  is  necessary,  first,  according  to  what  has  been 
laid  down  in  (169),  to  convert  into  cubic  yards  the  two  pieces 
of  work;  the  one  already  executed,  and  the  other  to  be  per- 
formed. This  we  do  by  multiplying  together  the  length,  breadth, 
and  depth  in  each  case.  Besides,  since,  according  to  the  enun- 
ciation, the  earth  of  the  second  is  three  times  more  difficult  to 
excavate  than  the  first,  if  we  express  by  1  and  3  the  relative 
difficulties,  we  must  introduce  into  the  two  products,  of  which 
we  have  just  spoken,  the  factors  1  and  3. 

This  established,  after  having  placed,  as  in  the  preceding  pro- 
blem, all  the  numbers  comprised  in  the  enunciation  upon  two 
difi'erent  lines,  we  are  led,  by  a  course  of  reasoning  entirely 
similar  to  that  which  we  pursued  in  the  solution  of  the  third 
problem,  to  form  two  new  lines  representing, —  the  one,  the  work 
done  by  1  man  in  one  hour  and  in  one  day ;  the  other,  the  work 
done  by  1  man  in  one  hour  and  in  x  days. 

Now,  it  is  clear,  that  these  two  quantities  of  work  must  bear 
to  each  other  the  direct  proportion  of  the  two  periods  employed 
to  perform  them.  We  have  then  the  equality  (1)  given  in  the 
table  of  the  calculations,  whence  we  deduce  the  final  equation 
there  given ;  and,  effecting  all  the  operations  indicated,  this  equa- 
tion gives,  finally, 

a:  =  5493VT; 

that  is  to  say,  it  would  require  549  days,  and  ^^j,  or  about  J  of 
a  day,  for  860  men  to  excavate  the  second  canal. 

173.  The  problems  which  precede,  suffice  to  exhibit  the  steps 
to  be  followed  when  the  method  of  Reduction  to  Unity  is  em- 
ployed. 

But  it  may  be  useful,  perhaps,  to  consider  the  results  furnished 
by  the  last  two  problems,  in  order  to  deduce  from  them  some  new 
consequences  concerning  the  use  of  direct  and  inverse  ratios. 
17* 


198  THEORY    OF    RATIOS    AND    PROPORTION. 

The  analysis  of  the  problem  in  (171)  led  to  an  expression  for 
the  number  of  cubic  yards  sought, 

1800x5x8x960 


7x4x500 


Now,  if  we  go  back  to  the  enunciation  of  the  question,  in 
order  to  distinguish  the  correspondents  of  each  species,  and  if 
we  separate  by  means  of  the  sign  of  multiplication  (X)  the  dif- 
ferent ratios  of  each  term  and  its  correspondent,  we  shall  be  able 
to  place  the  preceding  expression  under  the  form 

X         ,       ^       960 
1800  ~  4  -^  ^  "^  500  ' 


or  again,  under  this, 


X         4       960 


1800      I      500 


Examining  the  product  in  the  second  member,  we  see  that  the 
second  factor,  which  is  the  ratio  of  the  two  numbers  of  men  to 
be  clothed,  is  direct  with  that  of  the  numbers  of  yards  of  cloth, 

X 

;  while  the  first  factor,  or  the  ratio  of  the  two  breadths,  is 

X 

inverse  with  the  same  ratio,  ;   thus,  this  last  ratio,  called 

loOO 

compound  (168),  is  equal  to  the  product  of  the  ratios  of  the  two 

numbers  of  men,  and  of  the  two  breadths,  direct  for  the  men, 

and  inverse  for  the  breadths.     And,  in  fact,  the  more  men  there 

are  to  clothe,  the  more  cloth  necessary ;  but,  the  wider  the  cloth, 

the  smaller  number  of  yards  necessary  to  make  a  given  quantity. 

The  expression  obtained  in  the  problem  of  (172), 

2900x12x5x3x500x12x57 
^^      860x10x1800x7x3x1      ' 

can  be  put  under  the  form 

X        2900       .,       5       3  V  ^^^  V  ^2 

57T800  ^  ^  ^  -'^  •     seo""  10' 


THEORY   OF   RATIOS   AND   PROPORTION.  199 

and  we  see  also,  in  this  case,  that  the  ratio  of  the  two  numbers 
of  days  necessary  for  the  performance  of  the  two  pieces  of  work 
is  equal  to  the  product  of  the  ratios  of  the  correspondents  of 
each  species;  direct  in  the  case  of  the  dimensions  of  the  canals 
and  the  difficulties  of  the  excavation;  but  inverse  for  the  num- 
bers of  workmen  employed,  and  the  numbers  of  hours  per  diem 
which  they  laboured.  • 

Whence  we  can  give  this  sort  of  General  Rule  for  the  resolu- 
tion of  every  question  whose  enunciation  contains  proportional 
quantities : 

Form  a  product  of  all  the  ratios^  direct  or  inverse,  of  the 
correspondents  of  each  species,  excepting  the  ratio  of  which  the 
quantity  sought  forms  one  part ;  then  equal  this  product  to  the 
ratio  of  the  quantity  sought  to  the  quantity  of  the  same  species 
with  itself 

We  obtain  thus  the  expression  of  the  equality  of  two  ratios, 
from  which  we  easily  deduce  the  value  of  the  unknown. 

Rule  of  Simple  Interest. 

174.  The  Simple  Interest  on  a  sum  of  money  is  the  profit 
arising  from  the  loan  of  this  sum  for  a  certain  time. 

The  sum  lent,  or  placed  out  at  interest,  is  called  the  Principal 
or  Capital. 

The  interest  upon  a  sum  of  money  depends  upon  the  amount 
of  the  Principal,  upon  the  time  for  which  it  is  lent,  and  upon 
what  is  called  the  rate  of  interest,  or  the  interest  which  a  certain 
fixed  sum  bears  for  a  given  fixed  period. 

Ordinarily,  the  rate  is,  in  the  United  States,  the  interest  which 
the  sum  of  one  hundred  dollars  bears  in  one  year,  and  hence  is 
called  the  rdiio, per  cent. 

This  rate,  which  we  consider  a  sort  of  unit  of  interest,  is 
purely  conventional,  and  depends  generally  on  the  abundance  or 
scarcity  of  capital.  Nevertheless,  there  are,  in  commerce  and 
banking,  certain  limits  (in  most  countries  fixed  by  law),  beyond 
which  the  rate  becomes  usury. 


200  THEORY   OF   RATIOS   AND   PROPORTION. 

It  is  evident  that  the  interest  on  two  principals  for  the  same 
period  must  be  proportional  to  the  principals,  (the  rate  being 
constant),  and  the  interest  on  the  same  principal  for  two  different 
periods,  are  proportional  to  the  lengths  of  the  periods. 

Whence  it  follows,  that  the  rule  of  interest  is  only  a  particular 
case  of  the  Rule  of  Three. 

Thus,  the  questions  which  arise  under  it  can  be  treated  in  the 
same  manner  as  the  preceding. 

175.  Example.  —  Required^  the  Interest  on  $4500  for  2  year% 
and  5  months,  at  the  rate  ofWl  for  every  $100 ;  or,  hy  ahhrevia- 
tion^  at  the  rate  of  7  per  cent,  per  annum. 

This  enunciation  can  be  thus  rendered  :  $100  bring  $7  in  one 
year,  or  l2  months ;  how  much  ought  $4500  to  bring  in  2  years 
and  5  months,  or  29  months  ? 

The  numbers  can  be  thus  arranged  : 

100     12  months         7 
4500     29        '<  oc 


The  quantities. 


1       I    month 
11'^ 

"7 

and 


7 
100x12 

X 

4500x29* 


100x12         4500x29' 


express  each  what  one  dollar  brings  in  one  month,  apd  must 
therefore  be  equal,  and  we  have, 

X         ^ 7 

4500x29  ""100X12^ 
whence, 

4500  X  29  X  7  ^  15  X  29  X  7 
^~       100x12       ~  4 

Reducing  to  decimals, 

a:  =  $761-25, 


THEORY   OP  RATIOS   AND   PROPORTION.  201 

the  interest  on  $4500  for  2  years  and  5  months,  at  7  per  cent, 
per  annum. 

176.  Generally,  let  us  denote  the  principal  by  a,  the  time  by  tj 
the  rate  per  cent,  per  annum  by  i  and  by  g,  the  interest  on  the 
capital,  by  a.     We  shall  have, 

$100     1  year     1  dollar. 


a        t 

9 

1        1 

100  ^^*''''^ 

1        1 

aXt 

ThoH, 

9           ** 
ax<~100' 

and,  consequently, 

aXi 
9=  innX'- 

(1) 


The  time  t  can  be  a  fractional  number  of  the  unit,  year  haying 
for  denominator  the  number  of  months  or  of  days  in  the  year. 
If  we  place  (1)  under  the  form 


aXi    ^ 


100 


it  can  be  translated  into  the  following  rule : 

In  order  to  determine  the  interest  (/,  multiply  the  given  prin- 
cipal hy  the  rate  of  interest  for  one  year,  and  divide  the  product 
hy  100;  then  multiply  the  result  hy  the  number  of  years,  frac- 
tional or  entire. 

Example.  —  Required,  the  interest  on  $2524  65,  at  4^  per 
cent,  per  annum  for  2  years  and  7  months. 


202        THEORY  OF  RATIOS  AND  PROPORTION. 

We  have,  first, 

2524-65  x4-5  =  11360-925. 
Dividing  by  100,  113-60925 


For  two  years, 


2  yrs.  7  mos. 
21^7-21850 


6  months,  56-804625 

1  month,  9.467437 


293-490562;  or,  $293-49. 

It  is  obvious  that  this  division  by  100  can  be  performed  on 
the  rate  before  the  first  multiplication,  thus  converting  that  into 
a  decimal  fraction,  by  which  the  principal  is  to  be  multiplied. 

Example. — Required j  the  interest  on  $365-874,  at  5-^  per 
cent,  for  one  year  ? 

This  rate,  5 J  per  cent.,  divided  by  100,  gives  0-055.  We 
then  multiply  365-874  by  0-055. 

365-874 
•055 


1829370 
1829370 


$20-12307    $20-12.  Ans. 

177.  This  second  method,  which  we  have  applied  in  the  last 
two  examples,  is  always  to  be  preferred,  especially  when  we  wish 
to  determine  the  interest  for  a  certain  number  of  days. 

Required,  for  example,  to  find  the  interest  on  $1748-19,  for 
113  days,  at  4|  per  cent,  per  annum.  (We  suppose  the  year  to 
contain  360  days,  30  days  for  each  month). 

We  multiply  1748-19  by  4|,  divide  by  100;  we  then  divide 
113  into  60  +  30  +  20+3  days,  and  find  the  interest  for  each  one 
of  these  parts  separately.  Summing  these  parts,  we  have  the 
interest  required. 


THEORY   OF   RATIOS   AND   PROPORTION.  -Oo 


Table 

of  Calculationi 
1748-19 

6992-76 
I  874-095 
\  437-0475 

8303-9025 

;. 

by  100, 

83-039025 

for  CDC  year's  interest. 

For  60  days, 

13-839837 

"  30     *^ 

6-919918 

half  of  the  above. 

"  20     " 

4-613279 

\               (C              it 

a     3     a 

0-691992 

J^  of  the  int.  for  30. 

26-065026 
Thus,  the  interest  on  $1748-19  for  113  days,  is  $26-06.* 

178.  The  equation  (1)  of  (176),  contains  the  solutions  of  four 
different  problems. 

1st.  Knowing  the  Principal,  time  and  rate,  to  find  the  In- 
terest. 

This  we  have  discussed  in  several  examples. 

2d.  Knowing  the  Interest,  time,  and  rate,  to  find  the  Principal. 

3d.  Knowing  the  Interest,  Principal,  and  time,  to  find  the  rate. 

4th.  Knowing  the  Principal,  Interest,  and  rate,  to  find  the 
time. 

All  these  admit  readily  of  solution ;  but  we  will  limit  ourselves 
here  to  an  example  of  the  fourth  problem,  treating  it  by  both  of 
the  methods  explained  in  a  preceding  article. 

*  The  rjito  of  6  per  cent,  per  annum  admits  of  the  following  abbrevia- 
tion of  the  above  rules  when  applied  to  a  given  number  of  months ;  6 
per  cent,  per  annum  is  J  per  cent,  per  month,  or  1  per  cent,  for  two 
month?.  Then  we  can  say,  in  order  to  find  the  interest  on  a  certain 
principal  for  a  given  number  of  months,  at  the  rate  of  6  per  cent,  per 
annum,  we  multiply  the  principal  by  J  the  number  of  months,  and  divide  by 
100. 


204  THEORY  OP  RATIOS   AND   PROPORTION. 

A  sum  of  $2524-65  brought  $293-49,  at  the-rate  of  4^  per 
cent,  per  an  num.  Required  the  length  of  time  the  sum  was 
placed  at  interest  ? 

First  Mode  of  Proceeding. 
100         1  year         4J 
2524-65      t  293-49 


1  1 

1  t 


4-50 

100 

293-49 

2524-65 


L  ~  ^Q^'^^       12?  __       29349        _  29349000 
r  ~  2524-65  ^  4T  ■"252-465x45  "  11360925' 

Effecting  the  division,  we  obtain  2  years  and  7  months,  ne- 
glecting a  fraction  less  than  0-001  of  a  month. 

Second  Method. 

2524-65 
4-1 


10098-60 
1262-325 


or,  dividing  by  100,       11360-925 

113-60925     interest  for  one  year. 

And  as  $293-49  is  the  interest  for  t  years,  we  must  divide 
29349000  by  11360925,  in  order  to  obtain  the  time  required,  t. 

Rule  of  Discount. 

179.  Discount  is  the  deduction  which  is  made  from  an  amount 
payable  at  the  end  of  a  certain  timCj  when  we  wish  to  mahe  it 
payable  at  the  present  time,  or  before  it  falls  due  by  agreement. 
It  is  usually,  in  bankers'  terms,  the  deduction  which  we  make 
from  the  face  (amount  of  a  promissory  note,  in  order  to  get  its 
cash  value.     This  reduction  is  usually  made  at  so  much  in  the 


THEORY   OF  RATIOS   AND   PROPORTION.  205 

hundred  per  annum;  and  this  is  the  rate  per  cent,  of  discount. 
The  discounter  is  he  who  cashes  the  note  by  anticipation. 

It  is  easy  to  see  that  the  rule  of  discount  is  the  same  with  the 
rule  of  interest,  with  this  diflference,  that,  in  the  latter  case,  the 
horrower  is  obliged  to  restore  to  the  lender  the  sum  lent,  in- 
creased by  its  interest ;  while,  in  the  case  of  discount,  the  pos- 
sessor or  maker  of  the  note  receives  only  the  diflference  between 
the  amount  of  the  note  and  the  discount  which  is  made  by  reason 
of  the  anticipation  of  its  payment. 

Example  First.  — Required^  the  discount  on  a  note  q/'$875'49, 
payable  in  18  months,  at  the  rate  of  4l'%  per  cent,  per  annum. 

First  Method. 

$100         12  months        4-80 
875-49   18        "  X 


^  1        "  Torm    <^is^o^°*  0^  ^1  ^^^  1  y^*^- 


1  1 

Then, 


X 


875-49x18 
4-80 


875-49x18      1200' 

4-80  X  875-49  x  18      40  x  87549  x  18 
whence,     x  = j^OO ==        1000000        > 

or,  performing  the  calculations, 

a:  =  63035280  =  63-04. 

Amount  of  the  note,  $875*49. 
Discount,    ....     63-04. 

Difference,    .     .     .     $812-45,    the   amount   which 
the  discounter  pays. 

18 


206        THEORY  OF  RATIOS  AND  PROPORTION. 

Second  Method. 
Amount  of  note,  $875-49 

Rate  of  discount  per  an.  4-8 

"700392 
850196 


4202-352 


dividing  by  100,  42-02352 
1  year,  6  months. 

1  year,  42-02352 

6  months,  2101176 

63-03528  as  above. 
This  example  suffices  to  show  the  identity  of  the  calculations 
under  the  Rules  of  Interest  and  Discount. 

Example  Second.  —  Required ,  the  discount  on  a  note  of 
$3478-19,  payable  in  286  days,  the  rate  being  6*25  per  cent,  for 
360  days. 

We  commence  by  decomposing  the  number  286  into  its  parts, 

180+90  +  10  +  5  +  1. 
We  then  make  the  following  table  of  calculations : 

347819 
6-25 


1739095 
695638 
2086914 


217-386875  discount  for  360  days. 


108-693437 

54-346719 

6-038524 

3-019262 

0-603852 


180 

90 

10 

5 

1 


172-701794  discount  for  286  days. 
$347819 
172-70 


$3306-49  cash  value. 


THEORY   OF  RATIOS   AND   PROPORTION.  207 

180.  The  generalization  of  the  rule  of  discount  would  lead  to 
the  equation 

(^)   ^  =  -100-' 

in  which  e  E  would  designate  the  discounts  on  $100,  and  on  the 
amount  of  the  note  respectively.  These  letters  would  simply 
replace  ff  and  t  of  (176). 

We  could,  according  to  the  equation  (2),  establish  the  enun- 
ciations of  four  general  problems  analogous  to  those  of  (178). 

181.  There  is  another  rule  of  discount  which  we  cannot  pass 
by ;  for  although  it  is  not  generally  employed,  it  appears  more 
rational  and  more  just. 

One  example  will  suflfice  to  give  an  idea  of  this  second  mode 
of  discounting. 

A  note  o/SlSOO,  'payable  at  the  end  of  15  montJiSj  is  presented 
to  a  hanker  J  who  agrees  to  cash  it  at  a  discount  of  4  6  per  cent, 
per  annum.  Required j  what  the  holder  of  the  note  must  re- 
ceive  f 

Analysis. — Admit,  that  4*60,  the  rate  of  discount,  is  at  the 
same  time  the  rate  of  interest  of  a  sum  put  out  at  interest. 

It  is  clear  that  the  possessor  of  the  note  ought  to  receive  now 
a  sum  which,  placed  at  interest  at  the  rate  of  4-6  per  cent,  per 
annum  for  15  months,  would  give  him,  capital  and  interest  added, 
the  amount  of  his  note. 

Now,  the  interest  of  $100  for  one  year,  being  4-60,  becomes, 
for  15  months,  4-60  +  \  of  $4-60,  or  $5-75. 

This  proves  that  $100,  placed  out  at  interest,  would,  at  the 
end  of  15  months,  become  $105-75,  capital  and  interest. 

Consequently,  $105*75,  payable  in  15  months,  are  equivalent 
to  $100  payable  now ;  then  $1,  payable  in  15  months,  is  equal  to 

,  payable  now ;  and,  consequently,  $1500,  payable  in  15 
105*75 

months,  can  be  represented  by 

100  X  1500         15000000         ^i .  -,  q  .  o  n^r 
-10575-'  ''  -10575-'  ^^  $1418*43*97, 

payable  now. 


208  THEORY   OP   RATIOS    AND   PROPORTION. 

Whence  it  follows,  that  the  holder  of  the  note  ought  to  receive 
from  the  banker  a  sum  of  $1418-44. 

In  fact,  if  we  calculate  by  the  Rule  of  Interest,  what  $1418-44 
ought  to  bring  at  the  end  of  15  months,  at  the  rate  of  4-60  per 
cent,  per  annum,  we  obtain 

^  =  81-5603, 
which,  added  to  1418-4397, 

gives  $1500-0000,  the  amount  of  the  note. 

Now,  instead  of  following  this  method,  the  banker  determines 
the  interest  on  $1500  for  15  months,  at  4-6  per  cent.,  which  gives 

$86-25; 
and  this  he  subtracts  from  $1500-00, 

$1413-75,   the   difference  which  he 
gives  the  possessor  of  the  note. 

N.  B.  It  is  to  be  remarked,  that  the  excess  of  $86-25  over 
$81-56,  or  $4-69,  which  the  banker  gains  by  the  last  operation, 
is  nothing  more  than  the  interest  on  $81-56  for  15  months.  For, 
multiplying  $81-56  by  5-75,  (rate  for  15  months,)  and  dividing 
by  100,  we  obtain  $4-6897,  or  $4-69. 

This  advantage  which  the  banker  gains,  independently  of  the 
profit  which  belongs  to  him  of  right,  is  a  sheer  loss  on  the  part 
of  the  holder  of  the  note. 

There  is  a  way  of  operating,  according  to  the  first  rule,  with- 
out injury  to  the  interests  of  the  possessors  of  notes.  This  would 
be  to  establish  a  rate  of  discount  a  little  lower  than  the  legal  rate 
of  interest ;  but  the  difiiculty  would  be  to  proportion  the  one  to 
the  other  fairly  under  all  circumstances. 

We  give  the  two  rules  or  enunciations  of  the  two  methods 
which  we  have  given  above. 

1st.  (179).  Calculate  the  interest  on  the  amount  named  in  the 
note  J  from  the  present  time  to  the  date  at  which  it  falls  due  ;  then 


THEORY   OF   RATIOS   AND   PROPORTION.  209 

subtract  this  interest  from  the  amount  named  in  the  note.      This 
will  be  the  cash  value  of  the  note. 

2d.  (181).  Find  what  $100,  placed  out  at  interest  for  the 
(jiven  time  will  brine/,  capital  and  interest  added  ;  then  multiply 
the  amount  named  in  the  note  by  the  ratio  of  $100  to  this  sum  ; 
the  quotient  will  be  the  present  value  of  the  note. 

The  first  rule  is  genferally  received  in  commerce,  because  it  is 
more  expeditious  and  convenient  with  regard  to  the  calculations. 
It  is,  moreover,  a  matter  of  agreement  between  the  banker  and 
holder  of  the  note. 

The  Questions  of  Compound  Interest  and  Discount,  and  the 
subject  of  Annuities,  require  a  knowledge  of  the  use  of  Loga- 
rithms, in  order  to  be  thoroughly  discussed.  Hence,  we  pass 
them  by  here,  merely  adding,  that,  in  Compound  Interest,  the 
interest  is  added  to  the  principal  at  the  end  of  the  year,  or  period 
chosen  as  unit;  and  then  this  sum  is  regarded  as  a  new  principal, 
on  which  the  interest  is  calculated  for  the  given  period,  and  again 
added,  &c.,  &c. 

There  are  a  great  number  of  questions,  such  as  Insurances, 
Rents,  &c.,  &c.,  which  come  under  the  rule  of  per  centage,  but 
they  present  no  difficulty  to  the  student  who  understands  tho- 
roughly the  preceding  discussions  of  proportional  quantities. 
They  are  generally  given  in  full  in  the  Commercial  Arithmetics. 

Rule  of  Fellowship. 

182.  The  Rule  of  Fellowship  has  for  its  object, 
To  divide  among  several  persons  associated  in  a  partnership 
business  the  profit  or  loss  which  results  from  their  enterprise. 

It  is  generally  admitted,  (and  it  is  moreover  conformable  to 
equity,)  that  the  part  of  gain  or  loss  of  each  partner  is  —  1st, 
proportional  to  the  amount  of  capital  he  has  placed  in  the  busi- 
ness, when  the  times  are  equal ;  2d,  proportional  to  the  time 
when  the  amounts  invested  are  the  same. 
18* 


210        THEORY  OF  RATIOS  AND  PROPORTION. 

From  this  it  results  that,  for  different  capitals  and  different 
times,  the  parts  are  proportional  to  the  products  of  the  capital 
stocks  by  the  times;  since,  by  multiplying  the  stocks  by  the 
tiuies  respectively,  we  bring  them  back  to  amounts  invested  for 
the  same  time.  Thus,  the  question,  considered  under  the  most 
general  point  of  view,  is,  to  divide  a  given  number  into  parts 
directly  proportional  to  other  numbers  also  given. 

Problem  First.  —  Three  persons  are  associated  in  trade.  The 
first  puts  $15,000  in  the  common  stock;  the  second^  $22,540; 
and  the  third,  $25,600.  At  the  end  of  one  year,  the  profits  of 
the  enterprise  are  $12,000.  Required,  the  share  of  each  one  of 
the  partners  f 

Analysis.  —  The  sum  of  the  three  amounts  invested  in  trade 
being  $63,140,  we  reason  in  the  following  manner : 

$63,140  have  given  a  profit  of  $12,000 ;  then  $1  has  produced 

dollars  profit.     Then,  for 

15000  ....  we  have  1|?^  x  15000  =  '-^^^  =  2850-807. 
63140  6314 

....... ..  s^^-'-^r-""- 


11999-998. 

Thus,  the  first  person  must  receive  $2850-81;  the  second, 
$4383-81;  and  the  third,  $4865-38. 

And  these  three  sums,  added,  reproduce  the  total  gain, 
$12,000. 

Problem  Second.  —  A  capitalist  commences  an  enterprise  with 
a  stock  q/ $25,000.  Five  months  later,  a  second  capitalist  joins 
the  enterprise,  and  furnishes  an  additional  capital  of  $40,000. 
Six  months  after  this  first  addition,  a  third  capitalist  adds 
$60,000.      At  the  end  ff  two  years  the  partnership  is  dissolved, 


THEORY   OF   RATIOS   AND   PROPORTION.  211 

after  having  realised  a  profit  of  $76,000.     Required^  the  share 
of  each  partner? 

The  $76,000  are  to  be  divided  among  tlie  partners  proportion- 
ally to  the  products  of  their  respective  investments,  by  the  num- 
bers of  months  during  which  these  funds  were  in  the  enterprise. 
•  Now,  1st,  $25,000,  invested  for  24  months,  equal  25000  X  24, 
or  $600,000  vested  for  1  month;  2d,  $40,000  invested  for  19 
months,  are  equivalent  to  $760,000  for  1  month;  3d,  $60,000 
for  13  months,  are  equivalent  to  $780,000  invested  for  1  month. 
The  question  is  then  the  same  as  the  first.  Having  formed  the 
sum  of  the  three  amounts  invested  =  $2140000,  we  obtain  suc- 
cessively for  the  three  parts  or  shares  of  the  profit, 

First  share,       ^^^  X  600000  =  21308-411. 
'       2140000 

Second  share,    —jjr^^  X  760000  =  26990-654. 
'    2140000 

Third  share,      J^^^  X  780000  =  27700-934. 
2140000  

75999-999. 
The  shares  are,  respectively, 

$21308-42;  $26990-65;  $27700-93. 

183.  In  general,  let  it  be  required  to  divide  any  number,  a, 
into  parts  proportional  to  the  given  numbers,  m,  n,  p,  q  .  .  .  . 

Form,  first,  the  sum  of  the  numbers,  m^  n^  p,  q  .  .  .  .  then, 
multiply  each  one  of  these  numbers  by  the  ratio 


m-f-n+p  +  q-j-.... 
We  obtain,  thus, 

a  X  m  a  xn  a Xp 


m.  -j-  n  -{-p  -^  .  .  .  /     m  +  n  +p  +  .  .  .  .     m  -\-  n  +  p  + ' 

fractions,  which  have  the  same  denominator,  and  are  necessarily 
in  the  direct  proportion  of  their  numerators,  or  because  of  the 
common  factor,  a,  in  the  direct  proportion  of  m,  n,  p,  q  .  .  .  . 


212  THEORY   OP   RATIOS   AND   PROPORTION. 

When  the  numbers,  m,  n^  p,  q  .  .  .  .  are  fractional,  we  com- 
mence by  reducing  them  to  the  same  denominator,  and  then  the 
question  becomes  the  same  as  the  preceding. 

Divide  360  into  four  parts,  proportional  to  the  numbers 

2      7       11       17 

These  fractions,  reduced  to  the  least  common  denominator,  be- 

COI^G  6  4      8  4      8  8      5  1 

•  "se?  "ggj  -ggj  -gg- 
Then,  the  four  parts  must  be  respectively  proportional  to  the 
numbers  64,  84,  88,  51. 

The  sum  of  these  numbers  being  287,  we  have,  successively, 


For  the  first  part. 

Ifo  X  64=   80-28. 

''       second, 

3 fi-o  X  84  =  105-37. 

"       third. 

36^  X  88  =  110-38. 

"       fourth, 

If?  X  5T==   63-97. 

36000. 

184.  The  following  questions  belong  also  to  the  same  rule : 

Prohlem  Third.  —  Required,  to  divide  a  sum  of  $36,000 
among  four  persons,  so  that  the  second  shall  have  twice  as  much 
as  the  first;  the  third  as  much  as  the  first  two  together;  the 
fourth  three  times  as  much  as  the  third. 

We  can  make  the  first  share  a  principal  unit,  with  which  we 
compare  the  rest.  Calling,  then,  the  first  part  1,  the  second 
part  will  be  2,  the  third  3,  and  the  fourth  9,  by  the  conditions 
of  the  question. 

The  question  is  then  to  divide  ^36,000  into  four  parts,  propor- 
tional to  the  numbers  1,  2,  3,  9.    We  obtain  for  the  four  parts, 

36000  ,      36000      ^,^^ 

First  part,      ^-j-^-^-^-^^  X  1  or -^^  =  2400 

Second  part,  X  2  "     =  4800 

Third  part,  ^^         X  3  "     =  7200 

Fourth  part,  -^^  X  9  ^'     =  21600 


THEORY   OF   RATIOS   AND   PROPORTION.  213 

Problem  Fourth.  —  A  person  leaves  $40,000,  to  he  divided 
among  four  heirs,  so  that  the  first  shall  have  |  of  the  whole  ;  the 
second  |;  the  third  | ;  the  fourth  |.  Required ,  the  share  of  each 
heir. 

If  the  sum  of  the  four  fractions  was  exactly  equal  to  1,  the 
conditions  of  the  bequest  would  be  fulfilled  by  taking  successively 
I,  |j  I,  and  \,  of  $40,000.  But,  if  we  reduce  these  fractions 
to  the  same  denominator,  we  find 

yo>  "5^5?  ^0?  "go? 
the  sum  of  which  is  greater  than  1.  Hence,  the  bequest  would 
be  more  than  absorbed  by  the  three  first  parts.  But  if  the 
$40,000  is  to  be  divided  proportionally  to  the  four  numbers, 
\,  ^,  I,  \,  we  would  simply  have  to  divide  it  into  parts  propor- 
tional to  the  numbers  15,  36,  40,  and  30,  the  same  as  the  pro- 
blem in  (183). 

185.  We  add  here  a  rule  which  has  for  its  object  to  determine 
the  relative  value  of  the  coins  of  two  countries,  knowing  the 
proportions  between  these  coins  and  those  of  other  countries. 
It  consists  in  reducing  to  a  single  proportion,  by  multiplication, 
several  given  proportions.  It  is  really  nothing  more  than  an 
application  of  the  rule  of  compound  fractions,  or  fractions  of 
fractions. 

A  single  example  will  suffice  to  give  an  idea  of  the  rule  and 
the  mode  of  applying  it. 

Example. 

48  francs     ...     are  equal  to  39  English  shillings. 
13  English  shillings  "  8  German  florins. 

50  German  florins  "  9  ducats  of  Hamburg. 

15  ducats  of  Hamburg      ^^  43  roubles,  Russian. 

How  many  Russian  roubles  are  equal  in  value  to  2500  francs  ? 

If  48  francs  are  worth  39  shillings,  then  1  franc  is  worth  || 
of  a  shilling.  In  the  same  manner,  if  13  shillings  are  worth  8 
florins,  1  shilling  is  worth  -f^  of  a  florin ;  and,  consequently,  1 


214  THEORY   OF   RATIOS   AND   PROPORTION. 

franc  is  worth  ||  of  j\  of  a  florin.  Again,  if.  50  florins  are 
worth  9  ducats,  then  1  florin  equals  -^^  of  a  ducat.  Continuing 
this  reasoning,  we  find  that 

2500  francs  =  2500  times  ||  of  j%  of  J^  of  f  |  of  a  rouble. 

rri,        or^nn  ^              39x8x9x43x  2500        , , 
Then,  2500  francs 48  X  13  x  50  X  15        ''''^^''' 


Rule  of  Alligation. 

186.  The  questions  which  come  under  this  rule  are  of  two 
sorts : 

We  may  either  wish  to  find  the  mean  value  of  several  sorts  of 
things  J  knowing  the  number  and  particular  value  of  each  sort, 
or  it  may  he  required  to  determine  the  quantities  of  several  sorts 
of  things  which  must  enter  into  a  mixture,  knowing  the  price  or 
value  of  each  sort,  and  the  price  or  total  value  of  the  mixture. 

We  will  discuss  only  the  questions  of  the  first  nature ;  the  se- 
cond belonging  to  the  province  of  algebra. 

Example   First.  —  A   wine   merchant   has  mixed  wines  of 
different  qualities,  viz.,  250  pints,  at  60  cents  the  pint;   180 
pints,  at  75  cents ;  and  200,  at  80  cents.     Required,  the  price 
of  one  pint  of  the  mixture  9 
We  observe,  first,  that 

250  pints,  at  60  cents,  bring  $150 
180  "  at  75  "  "  US5 
200      "      at  80      ('         "      $160 

$445 
Giving  $445  for  the  total  price  of  the  three  quantities  of    250 
wine  mixed.  180 

If,  now,  we  form  the  sum  630  of  the  three  numbers,  200 
250,  180,  and  200,  the  question  will  obviously  be  reduced  "7^ 
to  the  following  : 

630  pints  of  wine  cost  $445 ;  what  is  the  cost  of  each  pint  ? 

71  cents  is  the  price  required. 


THEORY   OP  RATIOS   AND  PROPORTION.  216 

General  Rule.  —  In  order  to  find  the  price  of  the  principal 
unit  of  a  mixtvre  —  1st.  Multiply  the  price  of  this  principal 
unit  of  each  sort  of  thing  hy  the  number  of  units  of  this  sort,  and 
add  all  the  products.  2d.  Sum  up  the  numbers  of  units  of  these 
different  sorts.  3d.  Divide  the  sum  of  the  products  or  the  total 
price  by  the  sum  of  the  numbers  of  units. 

Or,  more  briefly — .Find  the  total  price  of  the  mixture  by 
summing  up  the  prices  of  its  parts.  Then  divide  this  total  price 
by  the  number  of  principal  units  in  the  mixture.  We  thus  obtain 
the  price  of  one  principal  unit. 

Example  Second.  —  We  wish  to  melt  together  23  kilogrammes 
of  silver,  826  thousandths  fine  j  14  kilogrammes  910  thousandths 
fine ;  and  19  kilogrammes  845  thousandths  fine.  Required,  how 
many  thousandths  fine  the  mixture  will  be  ?  That  is,  how  many 
parts  of  pure  silver  each  1000  parts  of  the  new  coin  will  contain  ? 
(We  say  an  ingot  of  gold  or  silver  is  -^-q,  or  880  thousandths,  &c., 
fine,  when  -f^,  or  880  thousandths  of  it  is  pure  silver  or  gold.) 
It  results,  then,  from  the  enunciation,  that 

1st.  23  k.  at  -825  =  23  X  -825,  or  18-975  A;,  of  pure  silver. 
2d.    14  k.  at  -910  ==  14  X  -910,  or  12-740  k.  " 

3d.    19  k.  at  -845  =  19  X  -845,  or  16-055Ar.  " 

56  47-770  « 

Then,  the  56  kilogrammes  of  the  mixture  contain  47-770 
kilogrammes  of  pure  silver.    Thus,  the  fineness  of  the  new  ingot 

will  be  expressed  by  — ^^ — ,  or  0-853;  that  is,  it  is  853  thou- 
sandths fine. 

187.  Mean  or  average  values. — The  determination  of  the 
m^an  values  of  several  things  of  difi'erent  values,  is  a  particular 
case  of  the  rule  of  alligation  of  the  first  sort. 

We  call  the  mean  value  of  several  things  whose  particular 
values  are  already  known,  the  sum  of  the  values  of  these  things 
divided  by  their  number.  Thus,  in  the  case  of  two  things,  the 
mean  value  is  the  half  sum  of  the  values  of  these  things. 


216        THEORY  OF  RATIOS  AND  PROPORTION. 

Example  Third.  —  The  leDgth  of  a  park  was  measured  four 
dififerent  times.  The  first  measurement  gave  250-439  metres; 
the  second,  250-695  metres;  the  third,  249-750  metres;  finally, 
the  fourth,  251-158  metres.     Required,  the  length  of  the  park? 

As  none  of  the  measurements  agree,  it  is  clear  that  the  only- 
means  of  answering  the  question  is  to  find  the  average  or  mean 
value  of  all  these  measurements.  We  find  for  their  sum, 
1002-042 ;  dividing  this  result  by  4,  we  obtain  250-5105  metres 
for  the  mean. 

Problems  which j  without  depending  on  fixed  or  General  RuJeSy 
can  nevertheless  he  resolved  arithmetically. 

188.  In  the  preceding  questions,  the  methods  of  arriving  at 
the  required  solution  are  fixed  and  general ;  that  is  to  say,  sus- 
ceptible of  being  applied  to  all  questions  of  the  same  nature. 
But  an  infinite  number  can  be  proposed  which  come  only  in  part 
under  these  methods,  or  do  not  in  any  manner  depend  upon 
them.  In  these  cases,  algebra  alone  furnishes  sure  and  direct 
methods  of  resolution.  Nevertheless,  we  will  show  how  these 
sorts  of  questions  can  be  resolved  arithmetically.  We  have  seen, 
(154),  that,  in  order  to  analyse  or  resolve  a  problem,  we  must, 
hy  reflecting  upon  the  enunciation,  endeavour  to  discover  in  the 
relations  established  among  the  numbers  which  enter  it,  the  suc- 
cession of  operations  to  be  performed  upon  the  known  quantities, 
in  order  to  deduce  from  them  the  values  of  the  unknown. 

Problem  First  — Required,  a  number,  of  which  the  half, 
third,  fourth,  and  ^ths,  added  together,  form  the  number  575  ? 

We  commence  by  remarking  that,  to  take  the  ^,  \,  \,  and  |, 
of  any  number,  and  add  them  together,  is  the  same  thing  as 
multiplying  this  number  by  the  sum  of  the  fractions  ^,  \,  \,  and 
|,  or  by  ^4°.  Now,  since  the  product  of  the  number  sought 
by  y^^  must  be  equal  to  575,  it  results  from  the  definition  of 
division,  that  this  required  number  is  equal  to  the  quotient  of 
675  divided  by  y/  ;  and  consequently  equal  to  575  x  j\^g. 


r  VN 


IVfeli 


THEORY   OF   RATIOS   AND   PROPORTION.  217 

Performing  tlie  operations  indicated,  we  find  420  for  the 
number. 

Verification,  420 

The  half,  =~210 

One-third,  =  140 

One-fourth,  =  105 

One-seventh,  =    60 

One-seventh,  =    60 

Total,  ~575 

Problem  Second.  —  Required,  three  numbers  whose  sum  is 
equal  to  96,  and  such  that  the  second  exceeds  the  first  by  2,  and 
the  third  exceeds  the  sum  of  the  other  tioo  by  4. 

It  is  evident  that,  if  we  diminished  the  second  number  by  2, 
it  would  become  equal  to  the  first ;  and  that  if  we  diminished  the 
third  by  2  -j-  4,  or  by  6  units,  it  would  become  equal  to  double 
the  first;  thus,  the  sum  of  the  three  numbers  would  be,  after 
these  two  subtractions,  four  times  the  first  number. 

Now,  the  difierence  between  96  and  2  -f-  4  -f  2,  or  8,  is  88 ; 
whence,  we  see,  that  the  first  number  is 

equal  to  one-fourth  of 88  =  22 

Then  the  second  is      ....      22  +    2  =  24 
And  the  third 22  X  2  +    6  =  50 

Verification,  96 

Problem-  Third.  —  Three  workmen  are  employed  to  do  a  piece 
of  work;  the  first  could  do  it  alone  in  12  days,  working  10  hours 
a  day  ;  the  second  in  15  days,  working  6  hours  a  day  ;  the  third 
in  9  days,  icorking  8  hours  a  day.  Required,  \st.  In  what 
number  of  hours  the  three  men  working  together  can  do  the  work; 
2d.  What  part  of  it  each  one  will  do  ;  3d.  How  much  each  one 
ought  to  get  for  his  labour,  the  price  of  the  whole  work  being 
8108? 

Solution. — We  observe  that,  according  to  the  enunciation, 
the  first  workman  could  do  the  work  in  12  X  10,  or  120  hours ; 
then,  in  1  hour,  he  could  do  y^^  of  the  work.    The  second  could 
19 


218  THEORY   OF 'ratios   AND   PROPORTION. 

do  it  in  15  X  6,  or  90  hours;  thus,  in  one  hour,  he  could  do  -g^^ 
of  it.  The  third  would  do  it  in  9  x  8,  or  72  hours ;  then  in  one 
hour  he  would  do  77^  of  it.  These  three  workmen  labouring  to- 
gether would  then,  in  1  hour,  do 

750  +  iJo  +  72  =  -BE^  or  J^  of  the  work. 

Now,  if  in  one  hour  they  do  3^^  of  the  work,  they  would  do 
the  whole  in  30  hours. 

Again,  since  in  one  hour  the  first  workman  does  j|^,  in  30 
hours,  he  will  do  -f^^j  X  30,  or  |  =  j^^.  In  the  same  manner, 
the  second,  in  30  hours,  performs  -^q  x  30,  or  -i  =  j\.  Finally, 
the  third  does  t^^^xSO^  or  j\. 

Then,  to  find  the  amount  to  be  paid  to  each  man,  we  must 
divide  $108  into  parts  proportional  to  the  three  fractions,  -j^^,  j*^, 
y\,  or  the  three  numbers,  3,  4,  5 ;  which  gives  $27,  $36,  and 
$45,  for  the  respective  wages  of  the  labourers. 

Exercises. 

1.  A  vessel  has  provisions  for  only  19  days;  yet,  by  calcula- 
tions, 25  days  must  elapse  before  she  can  reach  a  port.  Required, 
how  much  the  ordinary  rations  must  be  reduced  ? 

2.  Twenty  workmen,  working  15  days,  10  hours  a  day,  exca- 
vated a  ditch  65  yards  long,  by  2-30  yards  wide,  and  -75  of  a 
yard  deep.  Required,  how  many  days  it  would  take  36  men, 
working  12  hours  a  day,  to  dig  a  ditch  200  yards  long,  by  3 
yards  wide,  by  1-25  yards  deep;  the  difficulty  of  excavating  the 
first  earth  being  to  that  of  the  second  as  3  to  4. 

3.  For  what  period  must  $3000  be  placed  out  at  interest  at  6 
per  cent,  per  annum,  in  order  to  bring  $1325-50  ? 

4.  What  is  the  rate  of  discount  on  a  note  of  $2500,  payable 
in  18  months,  for  which  the  sum  of  $1860-45  was  paid  in  cash? 

5.  Four  partners  invested  the  same  sum  in  an  enterprise; 
the  funds  of  the  first  were  in  the  business  for  8  months ;  the 


THEORY   OF   RATIOS   AND   PROPORTION.  219 

second  for  7  months;  the  third  for  10  months;  and  the  fourth 
for  1  year.  Divide  the  profit  of  $1800  proportionally  to  the  in- 
vestments augmented  by  the  interest  on  each,  at  the  rate  of  4 
per  cent,  per  annum  ? 

6.  We  wish  to  divide  $60,000  among  three  persons,  so  that 
the  second  shall  have  twice  as  much  as  the  first,  less  $2500 ;  that 
the  third  shall  have  three  times  as  rnuch  as  the  first,  less  $5000. 
What  is  the  share  of  each  person  ? 

7.  Two  pounds  of  copper,  at  45  cents ;  7  pounds  of  zinc,  at 
70  cents ;  9  pounds  of  antimony,  at  50  cents,  are  melted  toge- 
ther.    What  is  the  price  of  one  pound  of  the  allojr  ? 

8.  A  person  was  asked  how  much  money  he  had  in  his  purse. 
He  answered,  If  you  add  to  the  sum  which  I  have,  |,  f ,  and  | 
of  that  sum,  I  would  then  have  175  dollars.  What  sum  of 
money  has  he  ? 


EXAMPLES. 

For  the  convenience  of  teachers,  we  annex  the  following  ex- 
amples for  practice,  as  but  few  are  embodied  in  the  work  itself. 
These  are  chiefly  selected  from  difierent  practical  compilations  on 
arithmetic. 

Addition. 

Add  together,  1225,  3473,  7581,  9064,  and  6060.  Ans. 
Add  together,  3004,  523,  8710,  6345,  and  784.  Ans.  19366. 
Add  together,  7500,  234,  646,  and  19760.      Ans.  28140. 
Add  together,  182796,  143274,  32160,  47047.  Ans.  405277. 
Add  together,  66947,  46742,  and  132684.      Ans.  246373. 


Subtraction. 

16844 
9786 

103034 
69845 

5987432 
278459 

7058 

33189 

7896600 
5403257 

5403257 
4250268 

5789232 
410204 

Multiplication. 

1st.  Multiply  328  by  2. 
Multiply  745  by  3. 
Multiply  20508  by  5. 
Multiply  3605023  by  6. 
Multiply  9097030  by  9. 

2d.  Multiply  725  by  300. 
Multiply  35012  by  2000. 
Multiply  9120400  by  90. 
Multiply  4890000  by  36000. 

Ans.  756. 
Ans.  2235. 
Ans.  102540. 
Jlns. 

Ans. 

Ans.  217500. 
Ans.  70024000. 
Ans.  820836000. 
Ans. 

(220) 


DIVISION  —  VULGAR   FRACTIONS. 


221 


3d.    Multiply  793  by  345. 

Multiply  471493475  by  4395. 
Multiply  89999000  by  97770400. 
Multiply  17204774  by  125. 
Multiply  3768  by  4230. 
Multiply  9648  by  6137. 


Ans.  273585. 

Ans.  2072213822625. 

Ans.  . 

Ans.  2150596750. 

Ans. . 

Ans.  49561776. 


Division. 

1st.  Divide  3788  by  2. 

Divide  4736511  by  9. 

Divide  78920  by  5. 

Divide  364251  by  3. 

Divide  34300  by  7. 
2d.  Divide  1203033  by  3679. 

Divide  49561766  by  5137 

Divide  2150596750  by  125. 

Divide  71900715708  by  57149. 
Ans. 

Divide  78674  by  200. 

Divide  32500000  by  520. 

Divide  36000000  by  3600. 

Divide  27489000  by  350. 


Ans.  1894. 
Ans.  526279. 
Ans.  15784. 
Ans.  121417. 
Ans.  4900. 
Ans.  327. 
Ans.  9648. 
Ans.  17204774. 

1258127.  Rem.  15785. 
Ans.  393  +  74  Rem. 
Ans.  62500. 

Ans. . 

Ans.  7854. 


Vulgar  Fractions. 

Reduction  of  Vulgar  Fractions  to  a  Common  Denominator. 
Reduce  |  and  |  to  a  common  denominator. 

Ans.  3g,  ^g. 
Reduce  ^,  |,  and  |  to  a  common  denominator. 

^ns.  18,  If,  If. 
Reduce  -f^,  J,  ^j  and  |,  to  a  common  denominator. 

>4?)«    _63_0       18911      1800      1750 
^^^'    3T5(J'    3T50?    3T^UJ    3T5IJ- 

Reduce  y,  I,  j^^j,  and  /^,  to  a  common  denominator. 


Ai 


19 


222 


EXAMPLES. 


Finding  the  Least  Common  MultipU. 

Find  the  least  common  multiple  of  13,  12,  and  4. 

Ans.  156. 
What  is  the  least  common  multiple  of  11,  17,  19,  21,  and  7  ? 

Ans.  . 

Find  least  common  multiple  of  6,  9,  4,  14,  and  16. 

Ans.  1008. 
What  is  the  least  common  multiple  of  1,  2,  3,  4,  5,  6,  7,  8,  9  ? 

Ans.  2520. 

Reduction  of  Fractions  to  the  Least  Common  Denominator. 
Reduce  ^,  |,  |,  and  |,  to  the  least  common  denominator. 

Ay,(i      6        8        9       10 

Reduce  j^g,  ^^,  and  |,  to  the  least  common  denominator. 

J^o       72         60_     320 
,  ^Ai&.    3g^,    3gQ,    3g^. 

Reduce  4>  I?  I?  i?  il?  ^^^  hh  *^  *^^  ^^^'^^  common  denomina- 
tor An<i     16     3fi     40     4  2      33     34 
^"^-                                                                            ^"^-  48'   45'   48'   48'   4S'   45" 

Reduce  |,  |,  5'^,  |,  and  y^^j,  to  the  least  common  denominator. 

Ans. . 

Reduce  |,  |,  |,  /j,  |,  to  the  least  common  denominator. 

A71S. . 


Greatest  Common  Divisor. 

Find  the  greatest  common  divisor  of  24  and  36. 

Ans.  12 
Find  the  greatest  common  divisor  of  312  and  504. 

Ans.  24 
Find  the  greatest  common  divisor  of  9024  and  3760. 

Ans.  752, 
Find  the  greatest  common  divisor  of  4410  and  5670. 

Ans.  630 
Find  the  greatest  common  divisor  of  3775  and  1000. 

Ans. 

Find  the  greatest  common  divisor  of  101  and  859. 

Ans. 


VULGAR   FRACTIONS.  223 

Reduction  of  Fractions  to  their  Simplest  Terms. 

Reduce  -f^-g^  to  its  simplest  terms.  Ans. . 

Reduce  j|§  to  its  simplest  terms.  Ans.  |fi. 

Reduce  |||J  to  its  simplest  terms.  Ans.  j^. 

Reduce  if-i^^^  to  its  lowest  terms.  Ans.  ^/gy* 


Reduce  ff  to  its  lojvest  terms.  Ans. 

Reduce  |ij  to  its  lowest  terms.  Ans. 

Addition  of  Fractions. 

Add  I,  4,  and  |  together.  Ans.  Lo_6. 

Add  I,  I,  and  ^^  together.  Ans. 

Add  -^^j  ^y^j  1,  and  y  together.  Ans. 

Add  j4j,  |,  |,  and  3^^  together.  Ans. 


Add  j\,  I,  §,  and  |  together.  Ans.  \i^§. 

Conversion  of  Fractions  into  Mixed  or  Entire  NumherSj  and 
vice  versd. 

Reduce  |  to  its  equivalent  whole  number.  Ans. 

Reduce  -|  to  a  mixed  number.  Ans.  3^. 

Reduce  ^5  to  a  mixed  number.  Ans.  3|. 

Get  out  the  entire  part  of  i^-^.  Ans.  24^2^. 

Get  out  the  entire  number  in  y.  Ans.  12^. 

Find  the  entire  part  in  ^|f  <^.  Ans. . 

Reduce  \%^^  to  a  whole  or  mixed  number.  Ans. . 

Bring  144|  to  a  fractiopal  form.  Ans. 

Bring  47 1  to  a  fractional  form.  Ans. 

Reduce  ^Ij^  to  a  fractional  form.  Ans.  3_y. 

Add  Y,  I,  3f ,  4|  together.  Ans. 

Add  6/2,  2i,  4|  together.  Ans. 

Subtraction  of  Fractions. 

Subtract  \ -^  \  from  ^  -\-  \.  Ans. 

From  I  take  |.  Ans. 

From  5|  take  4|  +  |.  Ans. 

From  fi|  take  -^■^.  Ans. 


224  EXAMPLES. 

Multiplication  of  Fractions, 

1st.  Multiply  4  by  8. 
Multiply  I  by  5. 
Multiply  4^  by  3. 
Multiply  I  by  24. 

2d.   Multiply  7  by  J. 
Multiply  22  by  f 
Multiply  15  by  I . 

3d.    Multiply  I  by  I  by  f . 
Multiply  3f  by  4if 

Required  the  product  of  5,  |,  |,  |,  and  4. 
Required  the  product  of  4^,  |,  i,  and  18 1. 

Division  of  Fractions. 


1st.  Divide  |f  by  9. 
Divide  14  by  7. 
Divide  |i  by  37. 

2d.   Divide  10  by  f . 
Divide  7  by  f  |. 
Divide  28  by  jf. 
Divide  16  by  3%. 

3d.    Divide  |  by  f 
Divide  4^  by  2|. 
Divide  //,  by  tII^ 
Divide  371^  by  :j  j^. 

DENOMINATE    NUMBERS.  225 

Fractions  with  Fractional  Terms,  or  Complex  Fractions. 

41  _t.  5 
Reduce      |   '    ^  to  a  simple  fraction. 

4.7 1       51 

41  -1.    2  4 

From       ^!  ^.7 take ^ 


25  +  6^         I 
27|  4 

2 

3  "n 

Divide     i-±4by4|. 


Multiply  -^  by  ^ 


23 1 

"^4  3 

Fractions  of  Fractions  ;  or,  Compound  Fractions. 

Reduce  ^  of  |  of  |  to  a  single  fraction. 

Reduce  \  of  |  of  ^,  or  |  of  20  to  a  single  fraction. 

Reduce  -\  of  ^^  of  ^^  to  a  single  fraction. 

Add  \  of  \o  to  I  of  |. 

Multiply  i  of  I  by  |J  of  2. 

Approximate    Valuation  of  Fractions  hy  means  of  Fractions 
having  Smaller  Terms. 
Value  If  I  in  twelfths. 

Find  the  approximate  value  ot  ||f  in  ninths. 
Find  the  approximate  value  of  |||  in  eighths. 
Find  the  approximate  value  of  |||  in  tenths,  in  hundredths, 
in  thousands. 

Denominate  Numbers. 
deduction  of  Compound  Numbers. 
1st.  Reduce  59  lb.  13  dwf.  5  gr.,  to  grains. 
Reduce  £121  Os.  9^d.,  to  half  pence. 
Reduce  365  days,  bh.,  4§',  48",  to  seconds. 
Reduce  5  miles,  3  furlongs,  1  pole,  and  2  yards,  to  yards. 
Reduce  375  cwt.  2921b.  boz.,  to  ounces. 
Reduce  77  A.  1  R.  14  P.,  to  perches. 


226  EXAMPLES. 

2d.    Find  the  compound  number  of  pounds,  sHillings,  knd 

pence,  in  5900  pence. 
Find  the  number  of  tons,  cwts.,  &c.,  in  36325  Ihs. 
Convert  123200  yards  into  a  compound  number,  whose 

principal  unit  is  miles. 
How  many  pounds,  ounces,  &c.,  in  704121  grains? 
In  5927050  minutes,  how  many  weeks,  days,  &c.  ? 

3d.    Reduce  25  days,  3  hours,  5  minutes,  to  the  fraction  of  a 
year. 
Reduce  3  furlongs,  2  poles,  3  yards,  and  2  feet,  to  the 

fraction  of  a  mile. 
Reduce  10  Ibs.^  12  oz.,  to  the  fraction  of  a  cwt. 
Reduce  |  a  penny  to  the  fraction  of  a  pound. 
Reduce  4^  grains  to  the  fraction  of  a  pound  Troy. 

4th.  Convert  |  of  a  pound  into  shillings  and  pence. 
Convert  -^^  of  a  year  into  days,  hours,  &c. 
Convert  |  of  a  mile  into  a  compound  number  of  its  lower 

denominations. 
Convert  |  of  a  pound  Troy  into  oz.j  dwts.j  grs. 
Convert  -f^  of  a  cwt.  into  its  equivalent  compound  num- 
ber. 

Addition  of  Compound  Numbers. 
Form  the  following  sums  : 


(1) 

(2) 

(3) 

£. 

s. 

d. 

yds. 

ft. 

inches. 

lb. 

oz. 

drs. 

149 

14 

n 

5 

2 

H 

17 

15 

15 

37 

11 

n 

4 

1 

2| 

27 

14 

11 

69 

14 

7 

31 

0 

10| 

16 

13 

9 

64 

13 

10 

6 

1 

11 

70 

0 

0 

47 

14 

lOi 

51 

2 

5 

5 

6 

n 

Add  ^  £  and  |  of  a  shilling  together. 

Add  I  rwt.  and  ^  of  an  ounce  together. 

Add  ^  mile,  |  of  a  furlong,  |  of  a  yard  together. 


DENOMIIJATE   NUMBERS. 


227 


Subtraction  of  Compound  Numbers . 


(1) 

(2) 

(3) 

£. 

s. 

d. 

ft). 

oz. 

c^r. 

fur. 

rod. 

3^^. 

6 

3 

10 

125 

0 

10 

13 

34 

3| 

2 

10 

11 

27 

1 

15 

12 

39 

4 

From  2  £  take  10  pence. 
From  ^  yard  take  |  inch. 
From  I  of  a  lb.  take  2^  grains. 

Multiplication  of  Compound  Numbers. 

Multiply  £1  lis.  Qd.  by  5. 

Multiply  £1  17s.  Qd.  by  63. 

What  is  the  cost  of  9  cwt.  5  lbs.  of  sugar,  at  £1  lis.  bd.  per 

cwt.P 
What  is  the  cost  of  7  yc7s.  2  /I.  3  in.  of  cloth,  at  the  price  of 

£3  6s.  4:d.  per  yard  ? 
Multiply  5  feet  6  inches  by  10  feet  10  inches. 
Multiply  7  yds.  2  feet  3  inches  by  11  feet  10  inches. 
What  is  the  cost  of  |  yard  of  cloth,  at  |  £  per  yard  ? 
What  is  the  cost  of  2  lbs.  ^  oz.  of  a  commodity  which  costs 

2s.  ^d.  per  lb.  ? 


Division  of  Compound  Numbers. 

Divide  £69  lis.  M.  by  9. 

Divide  £28  2s.  \ld.  by  6. 

Divide  375  miles,  2  fur.,  7  poles,  2  yds.,  1  foot,  2  in.,  by  39. 

If  9^  yards  of  cloth  cost  £4  3s.  l\d.,  what  is  the  price  per 

yard  't 
A  man's  income  is  £140  a  year ;  what  is  it  per  diem  ? 
If  2 J  yards  of  cloth  cost  lO^s.,  what  will  |  of  a  yard  cost? 
If  66  lbs.  of  sugar  cost  £4  2s.  Ad.,  what  is  the  price  per  lb.  ? 
Divide  126  square  feet  by  2  feet  10  inches. 


228  EXAMPLES. 

Decimal  Fractions. 

Numeration  of  Decimals. 

"Write  in  figures  the  following  numbers  :  — 

Eight,  and  two  thousand  seven  hundred  and  seven  ten  mil- 
lion ths. 

Forty-five,  and  seventeen  hundred  thousandths. 

One,  and  one  hundred  and  one  billionths. 

Twenty-four,  and  forty-five  ten  thousandths. 

Five  thousand  six  hundred  and  eighty-two,  and  two  ten  mil- 
lionths. 

Five  thousand  and  one  ten  millionths. 

Five  hundred  and  one  ten  thousandths. 

Addition  of  Decimals. 

Add  0-1257,  25700101,  3256-05,  22-056,  3-25,  2-207,  and 
0-002256  together. 

Add  00009,  1-0436,  3,  0-02,  and  -028  together. 

Add  3-0739,  5867,  000000201,  25-06,  0-6,  0-21,  1-75,  and 
•003  together. 

Add  28-29,  2-829,  0-2829,  311212105-6,  3112,  -121056, 
4-0003,  and  -01  together. 

Subtraction  of  Decimals. 

From  27-06,  subtract  2-05078. 

From  36-055,  take  0-072530. 

From  9,  take  -9,  -09,  -009,  and  -0009,  in  succession. 

From  10-00001,  take  0-11111112. 

From  27-854,  take  25-9999. 

Multiplication  of  Decimals, 

Multiply  -573005  by  -000754. 
Multiply  2-01013  by  24. 
Multiply  -356  by  12000. 
Multiply  -55  by  -55. 
Multiply  3  00001  by  -00002. 


DECIMAL   FRACTIONS.  229 

What  is  the  price  of  52-756  yds.  of  cloth,  at  10-06  shillings 
per  yard  ? 

What  is  the  compound  number  of  Ihs.  oz.,  &c.,  in  -625  of  a 
cwt.  ? 

How  many  shillings  and  pence  in  -3333  of  a  £  ? 

Convert  -076  of  a  mile  into  yards,  &c. 

Convert  -04678  of  .a  pound  avoirdupois  into  oz.  and  dr. 

Division  of  Decimals. 

Divide  11-8652  by  2-303. 

Divide  34-77421  by  1-03. 

Divide  -0100001  by  -01. 

Divide  22-0784  by  -002. 

Divide  475-28677  by  -4,  by  -04,  by  -004,  by  -0004,  by 
•00004. 

Divide  1572-36620  by  980. 

Write  the  respective  quotients  of  28-79  by  10,  100,  1000, 
10000,  100000,  -1,  -01,  -001,  -0001,  -00001. 

Divide  -1  by  -0001. 

Divide  9  by  -9,  by  -09,  by  -009,  by  -005,  by  -00012. 

Convert  -122  of  a  shilling  into  the  decimal  of  a  £.   . 

Convert  0  98  of  a  Ih.  avoirdupois  into  the  decimal  of  a  cwt. 

Divide  94-0369  by  81-022. 

Conversion  of   Vulgar  Fractions   into   Decimals,   and  some 

Miscellaneous  Examples. 
Reduce  |  to  decimals. 
Reduce  l|  to  decimals. 
Reduce  |£  to  decimals. 
Reduce  j||^  to  decimals. 
Divide  10  by  563  into  five  decimal  places. 
Convert  the  decimals  0*75,  0*25,  0-5,  and  0-225  to  their  sim- 
plest form  in  vulgar  fractions. 

Convert  10s.  Qd.  to  the  decimal  of  a  £. 
Convert  9^  months  into  the  decimal  of  a  year. 
Convert  17  hrs.  10  min.  25  sec.  to  the  decimal  of  a  day. 
What  is  the  compound  number  in  £5  75  ? 
20 


230 


EXAMPLES. 


Decimal  Denominate  Numbers. 


What  is  the  price  of  82-125  metres  of  cloth,  at  $6-76  per 
metre. 

Multiply  89-767  metres  by  2-25  metres. 

Divide  66-787  square  metres  by  10  375  metres. 

Convert  53-84  metres  into  feet. 

Convert  520-687  grammes  into  Ihs.  avoirdupois. 

Convert  15  feet  6  inches  into  metres. 

Convert  25  lbs.  6  oz.  avoirdupois  into  grammes. 

Convert  25°  36'  56"  of  the  sexagesimal  division  of  the  circle 
into  its  equivalent  in  the  centesimal. 

Convert  209°  Fahrenheit's  thermometer  into  its  equivalent  on 
the  Centigrade. 

Different  Systems  of  Numeration. 

Convert  325  and  422  of  the  decimal  system  into  their  equiva- 
lent numbers  in  the  nonary  system,  and  multiply  them  together 
in  that  system. 

Convert  101,  233,  22101,  of  the  quaternary  system,  into  their 
equivalent  numbers  in  the  system  whose  base  is  six. 

Multiply  3023  by  4012  in  the  quinary  system,  and  c6nvert 
the  result  into  its  equivalent  in  the  decimal  system. 

Add  10011,  1001110,  101101,  101111  of  the  binary  system, 
and  convert  the  result  into  its  equivalent  in  the  decimal  system. 

All  the  Divisors  of  a  Number. 

Find  all  the  divisors  of  2820.  Number  of  divisors,  24. 

Find  all  the  divisors  of  38088.         Number  of  divisors,  36. 
Find  the  divisors  of  1764. 

Its  factors  are  2^x3^x7^     No.  of  divisors^  27. 
Find  the  prime  divisors  of  1665.  Ans.  3^,  5,  37. 

Find  the  prime  divisors  of  56700.         Ans.  22x  3*X  5'X  7. 
Find  the  prime  divisors  of  122108        Ans.  2='x7*x89. 
Find  the  prime  divisors  of  3329-  Ans. 


RULE   OF   THREE.  231 

Greatest  Common  Divisor  by  Prime  Factors. 

Find  the  greatest  common  divisor  of  12321  and  64345  by 
prime  factors. 

Find  the  G.  C.  D.  of  3775  and  1000. 
Find  the  O.  C.  D.  of  24720  and  4155. 

Greatest  Common  Divisor  of  Several  Numbers. 

Find  the  Gr.  C.  D.  of  1260,  1512,  2016,  and  7350,  by  the 
method  of  prime  factors. 

Find  the  a.  C.  D.  of  492,  744,  and  1044. 
Find  the  G.  C.  D.  of  216,  408,  and  740. 

Periodical  or  Repeating  Decimals. 

Convert  the  vulgar  fraction  |  into  a  periodical  decimal. 

Convert  ^|  irto  a  periodical  decimal. 

Convert  |f  |  into  a  periodical  decimal. 

Convert  q^J^j  into  its  periodical  decimal. 

Find  the  generatrix  or  vulgar  fraction  corresponding  to  the 
decimal  0-99999 Ans.  1. 

Find  the  generatrix  of  the  repeating  decimal, 
0-012345679012345679 Ans.  ^\. 

Find  the  generatrix  of  0-987654320987654320  .... 

Ans.  |fl. 

Find  the  vulgar  fraction  corresponding  to  the  decimal, 
8-927783783 Ans.  VeW- 

Find  the  vulgar  fraction  corresponding  to  the  repeating  deci- 
mal, 0-36538461538461.  Ans.  4.f 

Rule  of  Three. 

If  I  of  a  yard  of  cloth  costs  10s.  Qd.,  how  many  yards  can  be 
bought  for  £13  15s.  Qd.  ? 

If  100  workmen  can  finish  a  piece  of  work  in  22  days,  how 
many  will  it  require  to  finish  the  same  work  in  4  days  ? 

If  10  cwf.  can  be  carried  54  miles  for  27  shillings,  how  many 
pounds  can  be  cariieil  20  miles  for  the  same  money  ? 


232  EXAMPLES. 

If  15  yards  of  stuff,  |  yard  wide,  cost  27s.  M,,  what  will  40 
yards  of  the  same  stuff  cost,  one  yard  wide  ? 

If  the  keeping  a  horse  costs  87A  cents  a  day,  what  will  it  cost 
to  keep  11  horses  for  one  year  ? 

A  man  breaks,  owing  $14,000-57,  his  property  amounting  to 
$7840-26.     How  much  will  his  creditors  receive  in  the  dollar  ? 

Compound  Proportion  —  Reduction  to  Unity. 

If  a  man  travels  150  miles  in  4  days,  travelling  12  hours  a 
day,  in  how  many  days,  travelling  11  hours  a  day,  can  he  travel 
375  miles? 

If  150  bushels  of  corn  feed  18  horses  75  days,  how  many 
days  will  87  bushels  feed  11  horses  ? 

If  250  men,  in  4  days,  working  10  hours  a  day,  dig  a  trench 
275  yards  long,  3  yards  wide,  and  2  yards  deep,  in  how  many 
days,  working  9  hours  a  day,  will  25  men  dig  a  trench  430  yards 
long,  4  wide,  and  3  deep  ? 

If  a  regiment  of  soldiers,  consisting  of  970  men,  consume  350 
bushels  of  wheat  in  4  months,  how  many  soldiers  will  consume 
1500  bushels  in  3  months,  at  the  same  rate  ? 

If  the  transportation  of  15  cwt.,  2  quarters,  72  miles,  cost 
$5-64,  what  will  the  transportation  of  5  cwt.,  3  qrs.,  112  miles, 
cost  ? 

Rule  of  Simple  Interest. 

What  is  the  interest  on  $8079-74,  for  5  years,  at  6  per  cent? 

What  is  the  interest  on  $3750,  at  4|  per  cent,  for  5J  years  ? 

What  is  the  interest  on  $3375,  for  5  months,  at  6  per  cent, 
per  annum  ? 

What  is  the  interest  on  $4500,  at  5  per  cent,  per  annum,  for 
280  days  ?  • 

AVhat  is  the  interest  on  $3195-54,  for  7  years,  6  months,  and 
22  days,  at  6  per  cent,  per  annum  ? 

What  is  the  interest  on  £104^  3s.,  for  3  J  years,  at  5  per 
cent.  ? 


PERCENTAGE  —  RULE   OF   DISCOUNT.  233 

What  is  the  interest  on  $5556-25,  for  2  years,  7  months,  21 
,  at  4-^-  per  cent,  per  annum  ? 

In  what  time  will  $500  bear  an  interest  of  $500,  at  6  per  cent, 
per  annum? 

What  must  be  the  rate  of  interest  in  order  that  a  sum  put  out 
at  interest  must  double  itself  in  16 1  years. 

What  sum  put  o\it  at  interest,  at  the  rate  of  6  per  cent,  per 
annum,  will  produce  $575  ? 

Percentage. 

(We  add  a  few  questions,  the  solution  of  which  is  a  simple 
application  of  the  rule  of  proportion.) 

A  man  invested  $12,000,  and  lost  64  per  cent,  of  it.  How 
much  had  he  left  ? 

Two  men  had  each  $500.  One  spends  12^  per  cent,  of  his 
money;  the  other  15  per  cent,  of  his.  How  many  more  dollars 
did  the  last  spend  than  the  first? 

A  merchant  laid  out  $250  as  follows  :  —  He  pays  25  per  cent, 
of  his  money  for  clothes;  30  per  cent,  of  what  is  left  for  sugar; 
12  per  cent,  of  what  is  then  left  for  calicoes.  How  much  had 
he  remaining  ? 

A  man  has  $750,  and  spends  $85.  What  per  cent,  of  his 
money  has  he  expended  ? 

Out  of  a  cask  of  500  gallons,  60  gallons  are  drawn.  What 
per  cent,  is  this  ? 

If  I  pay  $756-75  for  5  hogsheads  of  tobacco,  and  sell  them 
for  $965-25,  what  per  cent,  do  I  gain  on  the  purchase-money? 

Rule  of  Discount. 

(Examples  to  be  worked  either  by  the  usual  rule,  or  by  the 
accurate  rule  of  Art.  (181). 

A.  has  a  note  against  B.  for  $5746,  payable  in  4  months.    He 
gets  it  discounted  at  7  per  cent,  per  annum.     How  much  does 
he  receive  ? 
20* 


234  EXAMPLES. 

A  planter  sold  produce  to  tlie  amount  of  $12,57^,  payable  in  6 
months.  He  gets  his  note  discounted  at  6  per  cent,  per  annum. 
IIow  much  does  he  receive  ? 

For  what  amount  must  a  note  be  drawn,  payable  in  1  year,  3 
months,  and  5  days,  so  that,  when  discounted,  its  present  value 
at  7  per  cent,  per  annum  shall  be  ^507*27? 

What  is  the  present  value  of  $8250,  payable  as  follows :  — 
One-half  in  4  months;  one-third  in  6  months;  the  rest  in  9 
months  ',  the  rate  of  discount  being  6  per  cent,  per  annum  ? 

What  is  the  present  value  of  $50'00,  payable  in  9  months,  the 
rate  of  discount  being  4^  per  cent,  per  annum  ? 

I  bought  goods  for  87500  in  cash,  and  sold  them  for  $9000, 
payable  by  a  note  in  6  months.  What  will  be  my  gain,  if  I  dis- 
count the  note  at  6  per  cent,  per  annum  ? 

Rule  of  Fellowship. 

A.  and  B.  have  gained  by  trading,  $230.  A.  put  into  stock 
$300 ;  B.  $500.     What  is  each  one's  share  of  the  profit  ? 

A.  and  B.  have  a  joint  stock  of  $4200,  of  which  A.  owns 
$3600,  and  B.  $600 ;  they  gain  in  a  year  $2000.  What  is  each 
one's  share  of  the  profits  ? 

Three  merchants,  A.  B.  and  C,  freight  a  ship  with  4340 
tons  of  coal.  A.  puts  in  1350  tons;  B.  875  tons;  and  C.  the 
rest.  In  a  storm,  the  seamen  were  obliged  to  throw  500  tons 
overboard.     How  much  of  the  loss  must  each  merchant  sustain  ? 

A.  put  in  trade  $500  for  4  months,  and  B.  $600  for  5  months ; 
they  gained  $240.  Divide  it  between  them  in  the  compound 
ratio  of  the  times  and  capitals. 

Four  traders  form  a  company.  A  puts  in  $400  for  5  months ; 
B.  $700  for  8  months;  C.  $840  for  6  months;  D.  $1500  for  10 
months.  They  lose  $1000.  Divide  the  loss  in  the  composite 
ratio  of  the  times  and  sums  invested. 

A.  put  $1500  in  trade  for  15  months  with  B.,  who  put  in 
$1000  for  18  months.  They  gain  $800.  Divide  the  gain  in  the 
ratio  of  the  two  sums  invested,  increased  by  the  interest  for  the 
two  period^;  at  6  per  cent,  per  annum. 


SOME  GENERAL  QUESTIONS.  235 


Rule  of  Alligation. 

A  grocer  mixes  80  gallons  of  whiskey,  at  37A  cents,  with  10 
gallons  of  water,  costing  nothing.  What  is  the  price  of  one 
gallon  of  the  mixture. 

A  man  employed  500  workmen,  160  of  whom  receive  wages 
at  the  rate  of  $2  a  day ;  200  at  $1-75 ;  and  140  at  SI -50.  What 
is  the  average  per  diem  of  each  labourer  ? 

A  mixture  being  made  of  5  Ih.  of  tea,  at  6s.  per  lb.  ;  19  lb.  at 
10s.  Qd.  per  pound;  and  15  Z^.  at  4s.  9d.  per  pound.  What  is 
one  pound  of  it  worth  ? 

On  a  certain  day  the  thermometer  indicated  the  following 
temperatures  :  —  From  6  A.  M.  to  10  A.  M.,  65°  ;  from  10  A.  M. 
to  1  P.  M.,  76° ',  from  1  P.  M.  to  4  P.  M.,  87° ;  from  4  P.  M.  to 
6  P.  M.,  70°.     What  was  the  mean  temperature  of  the  day  ? 

Some  General  Questions. 

Divide  $2000  among  A.  B.  and  C,  so  that  B.  may  have  $100 
more  than  A.,  and  C.  $70  more  than  B. 

Find  two  numbers  such,  that  if  we  add  21  to  the  first,  the  re- 
sulting sum  shall  be  5  times  the  second  number;  and  if  we  add 
21  to  the  second,  the  resulting  sum  shall  be  three  times  the  first 
number. 

Two  men  are  travelling  on  the  same  road,  in.  the  same  direc- 
tion ;  the  first  is  50  miles  ahead  of  the  second.  The  first  travels 
25  miles  a  day ;  the  second  35  miles  a  day.  How  many  days 
must  elapse  before  the  second  shall  overtake  the  first  ? 

The  hour  and  the  minute  hands  of  a  clock  are  exactly  toge- 
ther, and  it  is  between  4  and  5  o'clock.  What  o'clock  is  it 
exactly  ? 

A  reservoir  of  water  has  two  cocks  to  supply  it ;  by  the  first 
alone  it  may  be  filled  in  40  minutes ;  by  the  second,  in  50  mi- 
nutes; and  it  has  a  discharging  cock  by  which  it  may,  when 


236  EXAMPLES. 

full,  be  emptied  in  25  minutes.  Now  these  three  cocks  being 
all  left  open,  in  what  time  will  the  cistern  be  filled  ? 

A  father  devised  1  of  his  estate  to  one  of  his  sons ;  |  of  the 
remainder  to  another ;  and  the  remainder  to  his  wife.  The  sons' 
legacies  differed  by  $500.     What  did  the  widow  receive  ? 

There  is  an  island  73  miles  in  circumference,  and  three  pedes- 
trians start  together,  to  travel  in  the  same  direction  around  it. 
A.  goes  5  miles  a  day;  B.  8;  and  C.  10.  In  what  time  will 
they  all  come  together  again  ? 

What  number  is  that  from  which,  if  you  take  |  of  |,  and  to 
the  remainder  add  ^^  of  -i^q,  the  sum  will  be  10 


.NOTES. 

Note  A 

We  have  spoken  in  (135)  of  the  numbers  in  any  system  whose 
base  is  h,  which  possess  analogous  properties  to  9  and  11  in  the 
decimal  system.  These  numbers  are  h  —  1  and  &  +  1,  and  the 
properties  which  they  enjoy  are  the  following : 

1st.  When  the  sum  of  the  figures  of  any  number  whatever  is 
divisible  by  &  —  1,  this  number  is  itself  divisible  by  h  —  1. 

2d.  Any  number  is  divisible  by  5  -f  1,  when  the  difference 
between  the  sum  of  the  figures  in  the  odd  places,  counting  from 
the  right,  and  the  sum  of  the  figures  in  the  even  places  is  0,  or 
a  multiple  of  i  +  1. 

3d.  The  remainder  of  the  division  of  a  number  by  each  one 
of  the  two  numbers,  h  —  1  and  h  -\-l,  is  obtained  in  the  first 
case  by  the  aid  of  the  sum  of  the  figures ;  and,  in  the  second 
case,  by  the  difference  between  the  two  sums  of  the  figures  of 
the  odd  places  and  those  of  the  even  places. 

We  have  indicated  in  (135)  how  these  properties  may  be 
proved. 

Algebra  furnishes  also  another  means  of  demonstrating  these 
properties,  founded  on  the  principles, — 

1st.  That  If  —  1  is  always  divisible  by  h  —  1 . 

2d.  That  If  —  1  is  divisible  by  h  +  1,  when  m  is  an  even 
number,  and  5™  +  1  is  divisible  by  i  -f-  1  when  m  is  odd. 

We  give  here  the  systems  of  numeration  of  the  Greeks  and 
Romans,  with  their  notation,  the  latter  being  still  used  to  indi- 
cate ordinal  numbers  at  the  beginning  of  Chapters,  Sections,  &c. 

(237) 


238  NOTES. 


In  the  Roman  Notation. 

One  was  written  with  a  single  mark I. 

Two  with  two  marks  11. 

Three with  three  marks Ill, 

Four nil. 

Five  was  written  IIIII. 

And  so  on,  to  ten. 

Ten  was  written  X. 

Two  tens XX. 

Three  tens ,...  XXX. 

And  so  on,  to  ten  tens,  the  intermediate  number  being  written 
by  combining  the  two  sets  of  characters.  This  being  too  cum- 
brous, instead  of  writing  five  marks  for  the  number  five,  they 
took  the  upper  half  of  the  ten  (X)  or  (Y),  to  express  it.  And 
also  the  convention  was  adopted  (in  addition  to  the  one  adopted 
above,  viz.,  that  like  characters  placed  together  indicate  that  the 
numbers  thereby  represented  are  to  be  added)  that  a  character 
representing  a  smaller  number,  placed  before  a  character  repre- 
senting a  larger  number,  indicated  that  the  first  was  to  be  sub- 
tracted from  the  second,  and  placed  after  it ;  or  on  the  right,  is 
to  be  added  to  it.  Thus,  Y  being  five,  |Y  would  be  four,  and 
Yl  six,  Yll  seven,  |X  nine,  XI  eleven,  &c. 

Instead  of  writing  ten  X's  for  one  hundred,  the  character  £ 
was  adopted ;  the  lower  half  of  this,  [,  represented  fifty.  Then, 
instead  of  writing  four  X's  for  forty,  it  is  written  XL ;  t^^at  is, 
fifty  less  ten ;  and  sixty  is  written  [X ;  seventy,  [XX ',  eighty, 
LXXXj  ^^^  ninety,  XC-  These  characters  were  afterwards  re- 
placed by  the  letters  which  they  resembled.  I  was  put  for  1 ; 
V  for  Y;  X  for  X;  L  for  L  J  C  for  £ ;  D  for  the  character  re- 
presenting five  hundred ;  and  M  for  one  thousand. 

The  fundamental  operations  upon  numbers  can  be  performed 
very  readily  by  the  Roman  notation,  though  the  notation  is  by 
no  means  so  simple  and  convenient  as  the  Arabic. 


Example.  —  Add 


NOTES.  239 

CCLXVIII. 

DCLXXIII. 

CXLVII. 


MLXXXVIII. 


It  is  obviously  most  convenient  to  begin  on  the  right.  Adding 
the  I's,  we  find  eight  of  them,  or  one  V  to  be  carried,  and  III 
remaining.  We  set  down  the  III,  and  add  the  V  to  those  in 
given  numbers.  We  thus  obtain  three  Vs,  or  one  X  and  one  V 
remaining,  which  we  set  down.  We  add  the  X  to  those  in  the 
given  numbers,  which  come  after  the  L's,  taking  care  to  subtract 
from  this  sum  according  to  the  convention  of  the  notation,  the 
X,  which,  in  the  third  number,  comes  before  L.  We  thus  obtain 
three  X^s,  which,  as  they  do  not  make  one  of  the  denomination 
of  L,  we  write  down  in  the  result.  Adding  the  L's  we  obtain 
one  C  and  one  L,  which  we  write  down.  Summing  up  the  C's, 
we  get  five  C's  or  one  D,  which  we  carry  to  the  next  column ; 
and,  by  adding  the  D's,  we  get  M,  or  ten  hundreds.  The 
result  is  then  MLXXXVIII.  And  so  on  for  the  other  opera- 
tions. 

The  Grreeks  used  their  letters  in  several  different  ways  to  denote 
the  different  numbers.  The  most  general  system  of  notation  was 
the  following :  —  To  express  the  9  units,  9  tens  and  9  hundreds. 
They  divided  the  alphabet  into  three  parts ;  but,  as  the  alphabet 
contained  only  24  letters,  three  new  signs  were  introduced,  <;'  for 
six,  ^  for  90,  and  9  for  900.  All  the  numbers  less  than  1000 
were  denoted  by  these  letters  and  signs,  with  a  small  mark  a 
little  to  the  right  above  them.  A  similar  mark  under  the  letter 
represented  thousands.  Placing  one  letter  after  another  indicates 
that  they  are  to  be  added  together.  Thus,  a'===l,  ^'  =  10,  )3'  =  2, 
a^  =  1000,  t^  =  10-000,  ^',=  2000,  c'a'=ll,  t'|3'  =  12,  x'=20, 
xV  =  21,  x^  =  20,000,  p'=  100,  p,  =  100,000,  &c.,  &c.,  &c. 


240 


NOTES. 


Note  B. 

Abridged  Methods  of  Multiplication  and  Division. 

Questions  often  arise  wliicli  require  the  multiplication  of  two 
numbers  containing  a  considerable  number  of  decimal  figures, 
while  we  wish  only  to  regard  a  small  number  of  the  decimal 
figures  of  the  product.  It  is  important,  then,  to  have  a  method 
of  obtaining  the  product  of  any  two  decimals  with  the  degree  of 
approximation  which  the  enunciation  of  the  question  requires, 
without  being  obliged  to  calculate  all  the  partial  products  which 
the  usual  mode  of  multiplication  renders  necessary. 

This  method  is  the  abridged  method,  which  we  will  now  ex- 
plain. 

Let  it  be  required,  for  example,  to  obtain  to  within  less  than 
one  thousandth  the  product  of  the  two  numbers 

84-0783647  and  72-46538. 

We  would  attain  the  end  proposed,  if  we  could  form  a  number 
which  should  contain  all  the  thousandths  and  units  of  higher 
order  contained  in  the  total  product.  This  we  accomplish  in  the 
following  manner : 


Operation  Proposed. 

Verification  of  the  Operation. 

84-078364^ 

72465380 

8356427 

;^46387048 

588548548 

579723040 

16815672 

28986152 

3363132 

1507257 

504468 

57972 

42035 

2173 

2520 

434 

672 

28 

6092-7704^ 

4 

6092-770^0 


NOTES.  241 

We  commence  by  placing  fhe  multiplier  under  the  multipli- 
cand in  an  inverse  order,  and  so  that  the  figure  2,  in  the  simple 
ttnits  place,  shall  fall  under  the  figure  of  hundred-thousandths ; 
that  is  to  say,  under  the  figure  holding  the  second  place  on  the 
right  of  that  whose  order  indicates  the  degree  of  approximation 
required :  then  the  figure  4  of  its  tenths  falls  under  the  figure  of 
ten-thousandths  of  the  multiplicand;  the  figure  6  of  its  hun- 
dredths under  the  figure  of  thousandths  ;  and  so  on  for  the  rest. 

By  this  arrangement  each  figure  of  the  multiplier  corresponds 
to  such  a  figure  of  the  multiplicand,  that  the  product  of  the  two 
gives  hundred-thousandths.  Thus,  the  figure  7  of  the  tens  of  the 
multiplier  corresponds  to  the  figure  of  millionths  of  the  multi- 
plicand, and  their  product  gives  hundred-thousandths.  In  the 
same  manner,  if  there  were  hundreds  in  the  multiplier,  the  figure 
should  be  placed  under  the  figure  of  ten-mill  ion  fhs  of  the  multi- 
plicand. 

Having  arranged  the  numbers  thus,  we  multiply  successively 
all  the  figures  ^f  the  multiplicand,  beginning  on  the  right  by 
each  one  of  the  figures  of  the  multiplier,  not  taking  any  account 
of  the  figures  of  the  multiplicand,  which  are  situated  on  the 
right  of  the  figure  by  which  we  are  multiplying;  and  we  place 
the  products  (considered  as  resulting  from  the  multiplication  of 
two  entire  numbers),  one  under  the  other,  so  that  their  simple 
units  shall  fall  under  each  other.  We  add  then  all  these  pro- 
ducts, and  separate  on  the  right  of  the  sum  five  decimal  figures, 
and  draw  a  mark  across  the  last  two. 

The  part  on  the  left  of  these  last  two  figures  is  the  required 
product. 

In  order  to  see  the  reason  of  this  mode  of  operating,  and  to 
convince  ourselves  that  we  obtain  thus  the  desired  degree  of 
approximation,  it  suffices  to  remark  that,  at  each  partial  multi- 
plication, we  neglect  several  hundred-thousandths,  the  summation 
of  which  gives  some  ten-thousandths  of  error.  But,  admitting  as 
an  average,  that  we  commit  an  error  of  5  hundred-thousandths 
at  each  partial  multiplication,  we  see  that  it  would  require  10 
partial  multiplications,  or  10  figures  in  the  multiplier,  in  order 
21 


242  NOTES. 

that  the  error  might  reach  50  hundred-thousandths,  or  5  ten- 
thousandths  ;  and  20  figures  in  the  multiplier,  in  order  that  the 
error  should  be  10  ten-thousandths,  or  one  thousaridth. 

Verification  of  the  Operation. 

In  order  to  verify  the  result  obtained,  it  is  best  to  pursue  the 
following  method  :  —  We  take  the  multiplier  for  the  multiplicand, 
and  reciprocally,  as  the  table  of  operation  shows ;  and  we  average 
the  new  multiplier  as  in  the  first  operation ;  we  then  perform  the 
partial  multiplications  in  the  same  manner,  except  some  modifica- 
tions, which  it  is  necessary  to  indicate. 

1st.  We  have  placed  a  0  on  the  right  of  the  new  multiplicand, 
in  order  that  the  last  figure,  8,  of  the  new  multiplier,  may  have 
its  correspondent. 

2d.  We  have  drawn  a  line  across  the  first  figure,  7,  on  the 
left  of  this  multiplier,  as  it  ought  not  to  give  any  partial  product 
according  to  the  rule  established  above. 

3d.  In  each  partial  multiplication,  we  take  care  to  add  to  the 
product  of  each  figure  of  the  multiplier,  by  the  figure  above 
which  corresponds  to  it  immediately,  the  figure  to  he  carried, 
which  is  the  product  of  this  same  figure  of  the  multiplier  by  the 
figure  which  is  on  its  right  in  the  multiplicand. 

Thus,  in  the  multiplication  by  the  4th  figure,  7,  of  the  multi- 
plier, we  have  added  to  the  product  of  5  by  7  the  2  units  which 
would  have  to  be  carried  up  from  the  product  of  the  3  imme- 
diately on  the  right  of  the  figure  5  by  the  same  figure  7. 

4th.  Finally,  arrived  at  the  figure,  7,  of  the  multiplier,  across 
which  we  have  drawn  a  line,  we  have  multiplied  it  mentally  by 
the  figure  7,  which  is  on  its  right  in  the  multiplicand,  and  we 
have  written  the  figure,  4,  to  be  carried,  of  this  mental  product, 
below  the  preceding  product. 

This  last  modification  offers  two  advantages :  the  first,  that  it 
lessens  much  the  errors  committed;  and  the  second,  that  it 
enables  us  to  judge  whether  it  is  necessary  to  alter  by  a  unit  the 
figure  at  which  the  approximation  stops,  in  order  to  obtain  a 
more  exact  result. 


NOTES. 


243 


la  the  example  just  discussed,  we  have  found  47  for  the  last 
two  figures  of  the  first  operation,  while  in  the  verification  we 
find  60.     All  the  other  figures  are  the  same  in  both  operations. 

Then,  6092-771  is  the  value  of  the  required  product  to  within 
less  than  a  thousandth. 

We  give  an  example  a  little  more  complicated.  Let  the  two 
numbers  be 

1307-510300896472  and  256-10978641, 

of  which  we  wish  to  obtain  the  product  to  within  less  than 
000001. 


Operation. 

1307-510300896^^ 
14687901652 


Yerijicati^. 

256-1097864100 
^^{^6980030157031 


2615020601792 

2561097864100 

653755150445 

768329359230 

78450618048 

17927685048 

1307510300 

1280548932 

117675927 

25610978 

9152570 

768329 

1046008 

2048 

78450 

230 

5228 

15 

130 

334866-18389^0 

334866-1838^^ 


The  result  required  is  here  334866-18389,  to  within  less  than 
0-00001. 

Example.  — Find  to  within  less  than  001  the  product  of  the 
two  numbers  89-91666  and  47-19. 

Remark.  —  The  method  can  be  applied  equally  well  to  the 
approximate  multiplication  of  two  entire  numbers. 


244 


NOTES. 


Example.  —  Required  the  product  of  4702564917  by  2305687, 
to  within  less  than  a  million. 

470256497-00 

7865032 


94051299400 

14107694910 

235128245 

28215384 

3762058 

329175 


1084264291^^  1084264292  millions. 

We  will  certainly  obtain  the  product  to  within  less  than  a 
million,  by  taking  account  of  the  hundreds  of  thousands  and  the 
tens  of  thousands;  that  is,  by  calculating  in  the  product  two 
figures  more  than  the  number  required.  In  order  to  do  this,  we 
arrange  the  multiplier  below  the  multiplicand  in  an  inverse  order, 
and  so  that  the  figure  7,  of  its  simple  units,  shall  fall  under  the 
figure  of  tens  of  thousands  in  the  multiplicand ;  then,  the  tens 
figure  8,  of  the  multiplier,  will  fall  under  the  thousands  figure 
of  the  multiplicand,  &c.  Nevertheless,  as  the  figures  in  the 
hundreds  of  thousands  and  millions  places  would  not  have 
corresponding  figures  in  the  multiplicand,  we  supply  their  places 
by  two  O's  annexed. 

We  can  also  employ  an  abridged  method  of  division,  when  the 
dividend  and  divisor  are  composed  of  a  great  number  of  figures. 
But,  as  this  method  requires,  in  order  to  be  thoroughly  discussed, 
developments  which  could  not  be  given  here,  we  will  limit  our- 
selves to  giving  an  idea  of  the  mode  of  operating.  We  commence 
by  remarking  that  the  process  for  finding  the  quotient  of  the 
division  of  two  decimal  fractions,  with  a  given  approximation, 
can  always  be  reduced  to  finding  the  quotient  of  the  division  of 
two  entire  numbers  to  within  less  than  unity. 

For,  let  it  be  proposed,  for  example,  to  find  the  quotient  of  the 


NOTES.  245 

division  of  1234*569  by  27-35894  to  within  less  than  0001. 
According  to  the  rule  for  the  division  of  decimals,  we  must  place 
two  zeros  on  the  right  of  the  dividend,  which  reduces  the  opera- 
tion to  the  division  of  the  two  numbers, 

123456900  and  2735894. 

Then,  as  we  wish  "to  obtain  the  quotient  with  three  decimal 
places,  we  place  three  new  zeros  on  the  right  of  the  dividend, 
and  perform  the  division,  taking  care  only  to  separate  three 
figures  on  the  right  of  the  quotient  for  decimals. 

The  question  is  then  to  find  the  quotient  of  123456900000  by 
2735894,  and  only  regarding  the  entire  part  of  the  quotient. 
We  are  then  led  to  explain  the  rule  to  be  followed  in  the  abridged 
division  of  two  entire  numbers. 

This  rule,  principally  founded  upon  the  fact  that,  according  to 
the  ordinary  method,  the  determination  of  each  one  of  the  figures 
of  the  quotient  most  commonly 'depends  only  on  the  first  two  or 
three  figures  on  the  left  of  the  dividend ;  and  the  first  two  or 
three  figures  on  the  left  of  the  divisors  can  be  thus  enunciated. 

Suppress  on  the  right  of  the  dividend  as  many  figures,  less 
two,  as  there  are  in  the  divisor ;  divide  then  the  part  on  the  left 
hy  the  divisor,  and  if  there  is  no  remainder,  annex  to  the  quotient 
as  many  zeros  as  you  have  suppressed  figures  in  the  divisor. 

But  if  there  is  a  remainder,  divide  this  remainder  by  the 
divisor,  with  the  last  figure  on  the  right  cut  off.  Nevertheless,  in 
the  multiplication  of  the  new  divisor  hy  the  figure  obtained  in  the 
quotient,  take  care  to  add  the  figure  to  be  carried,  which  the  pro- 
duct of  the  figure  cut  off  from  the  divisor  by  this  figure  of  the 
quotient  would  give. 

Divide  then  the  new  remainder  by  the  divisor,  with  its  last  two 
figures  cut  off,  and  proceed  as  before. 

Continue  these  successive  divisions,  suppressing  at  each  division 
a  new  figure  on  the  right  of  the  divisor,  and  stop  the  operation 
ichen  the  divisor  is  reduced  to  its  first  two  figures  on  the  left. 
21* 


246  '  NOTES. 

In  order  to  render  this  method  intelligible,  wO'give  both  the 
ordinary  and  abridged  method  in  the  table. 

Ordinary  Method. 


540347056789046 


26170115 
10919846 
"25604697 
~5265668 
~24792099 


2786459 


193918897 


25004270 
"27125984 
"20478536 
973323 


Abridged  Method. 


5403470567  I  89046 


26170115 
10919846 
"25604697 
526566 


1939  I  18897 


247920 

25004 

2713 

"206 

""12 


In  the  second  operation,  we  separate  five  figures  on  the  right 
of  the  dividend,  since  there  are  &even  in  the  divisor;  and  we 
divide  the  part  on  the  left  by  the  divisor,  which  gives  the  first 


NOTES.  247 

four  figures   of  the   quotient,  1939,   and   for  the   remainder, 
526566. 

This  done,  we  draw  a  line  across  the  last  figure,  9,  of  the 
divisor,  and  divide  526566  by  278465;  the  quotient  is  1,  by 
which  we  multiply  the  divisor,  adding  1  to  the  units  figure  of 
the  product  for  the  9  suppressed  in  the  divisor;  we  then  sub- 
tract the  product  from  the  remainder,  and  thus  obtain  the  new 
remainder,  247920.  Cutting  off  a  second  figure  of  the  divisor, 
we  divide  247920  by  27864,  and  subtract  from  the  dividend  the 
product  of  27864  by  the  quotient  8,  this  product  being  aug- 
mented by  the  4  to  he  carried,  which  the  multiplication  of  8  by 
the  figure  5,  which  we  have  cut  off,  would  give.  We  proceed  in 
the  same  manner,  until  we  get  for  the  total  quotient  the  number 
193918897,  the  same  result  which  the  ordinary  method  gives. 

Let  us  now  apply  the  process  to  two  decimal  fractions,  taking 
the  example  proposed  above,  to  find  the  quotient  to  within  less 
than  0-001. 

Ordinary  Method. 

123456900000    2785894 


14021140 


45124 


3416700 
6808060 
13362720 
2419144 


Abridged  Method. 


1234569  I  00000 


140212 


272^/^0^4 


45124 


3418 
~682 
~135 
~26 


248  NOTES, 

Having  first  reduced  the  operation  to  that  of  entire  numbers, 
we  remark,  that  as  the  seven  figures  of  the  dividend,  which  re- 
main after  the  suppression  of  as  many  figures,  less  two,  as  there 
are  in  the  divisor,  do  not  contain  the  divisor,  it  is  necessary  in 
the  commencement  to  cut  off  the  last  figure,  4,  of  the  dividend, 
and  divide  1234569  by  273589. 

We  then  continue  the  operation  until  we  obtain  45124,  on  the 
right  of  which  we  separate  three  figures  for  decimals.  This  gives 
45-124  for  the  required  quotient  to  within  less  than  0*001. 


THE    END. 


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I.  The  Origin,  Orthography,  Pronunciation,  and  Definitions  of  all 
words  acknowledged  by  the  most  eminent  Lexicographers  as  properly 
belonging  to  the  English  Language,  with  many  idioms  and  phrases 
from  the  Latin  and  other  foreign  tongues. 

IL  An  arrangement  of  Synonyms  under  the  leading  words  :  a  new 
and  characteristic  feature,  designed  to  facilitate  the  practice  of  correct 
and  elegant  composition,  and  one  not  found  in  any  other  work. 

in.  A  Synopsis  of  words  differently  pronounced  by  different  Ortho- 
epists,  including  Walker,  Perry,  Jameson,  Knowles,  Smart,  and 
Worcester. 

IV.  Walker's  Key  to  the  pronunciation  of  Greek,  Latin,  and 
Scripture  proper  names. 

V.  A  Vocabulary  of  Modera.  Geographical  Names,  with  their  pro- 
nunciation, by  J.  Thomas,  M.  D.,  Editor  of  "  Lippincoit's  Pronoun- 
cing Gazetteer." 


RECOMMENDATIONS. 

"It  Is  the  most  complete  work  of  tlie  kind  yet  published.  The  defiDi'tions  nr^ 
clcHr  and  concise,  presentinj;  briefly  thx  VHrious  meHiiiiijrs  and  shades  of  meaning 
belonging  to  each  word.  .  .  .  The  pronunciiition  ia  flsitifitnctorily  indieat>d,  and  in 
most  cases  the  8yuoiiyiu8  of  the  words  defined  are  added,  a  grw«t  advuntagt-  to 
persfins  engaged  in  literary  compositions  "  Leeds  Thnea. 

"  To  all  who  wish  for  the  most  ciirnplete,  clieap  and  jjortahle  Dictionary  at 
this  moment  existing  of  our  noble  language,  including  an  immense  nuiss  of  plii!u- 
logic  matter  —  copious  Vocabularies  of  Scriptural,  Mythologic,  and  GeoirrHphic 
names — we  can  cordially  recommend  the  volume  before  us."        Lundun  Alias. 

'•  The  meaning  of  every  Knglish  word  in  all  its  various  shades  is  given  in  this 
admirable  work,  and  it  contains,  moreovei-  a  Uictionary  of  Syninyms." 

/uondon  Observer. 

"  A  marvel  of  nccurncy,  neatness  imd  cheapness. ...  It  \fi  a  eontriliusion  of 
substantial  service,  not  only  to  our  times,  but  for  posterity."    Wcsleyan  Banner. 

J.  B.  Mri'lNCUTT  At  CO.,  Publit-hers,  I'hii.adelphia.      ) 


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